Integral Calculus - IB Math AA Question Bank

Question 1

SL • Paper 2

Easy

Calculator Permitted

The gradient of a curve is given by

\frac{dy}{dx}=6x^2-4x+\frac{3}{\sqrt{x}}, \quad x>0

The curve passes through the point $(4,18)$ .

A

Find $y$ in terms of $x$ .

[4]

Write your answer here...

0

Question 2

SL • Paper 1

Medium

Non Calculator

A function $f$ satisfies $f'(x)=3x^{\frac{1}{2}}-\dfrac{2}{x}+e^{2x-1}$ , for $x>0$ , and $f(1)=4$ .

A

Find $f(x)$ .

[5]

Write your answer here...

0

Question 3

SL • Paper 1

Medium

Non Calculator

The curves $y=4-x^2$ and $y=x+2$ enclose a finite region.

A

Find the $x$ -coordinates of the points of intersection of the two curves.

[2]

Write your answer here...

B

Find the area of the enclosed region.

[3]

Write your answer here...

0

Question 4

SL • Paper 1

Medium

Non Calculator

A

Evaluate $\displaystyle \int_0^{\frac{\pi}{2}} \sin x(2+\cos x)^3\,dx$ .

[4]

Write your answer here...

0

Question 5

SL • Paper 1

Medium

Non Calculator

A function $f$ satisfies $f''(x)=6x-4\sin x$ , $f'(0)=3$ and $f(0)=1$ .

A

Find $f'(x)$ .

[3]

Write your answer here...

B

Hence find $f(x)$ .

[2]

Write your answer here...

0

Question 6

HL • Paper 1

Medium

Non Calculator

A

Evaluate $\displaystyle \int_0^1 xe^{2x}\,dx$ .

[4]

Write your answer here...

0

Question 7

SL • Paper 2

Medium

Calculator Permitted

The function $f$ satisfies

f'(x)=4\cos(2x-1)-\frac{3}{x+2}, \quad x>-2

It is given that $f(1)=5$ .

A

Find an expression for $f(x)$ .

[4]

Write your answer here...

B

Find $f(3)$ .

[2]

Write your answer here...

0

Question 8

SL • Paper 2

Medium

Calculator Permitted

For $0\le x\le 4$ , the shaded region is bounded by the curve

y=3\sin(0.6x)+2

and the $x$ -axis.

A

Explain why the area of the shaded region is equal to $\int_0^4 \left(3\sin(0.6x)+2\right)\,dx$ .

[2]

Write your answer here...

B

Find the area of the shaded region.

[3]

Write your answer here...

0

Question 9

SL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=x^2-4$ .

A

Write down the $x$ -intercepts of the graph of $f$ .

[1]

Write your answer here...

B

Find the total area enclosed by the graph of $f$ , the $x$ -axis and the lines $x=-1$ and $x=3$ .

[5]

Write your answer here...

0

Question 10

SL • Paper 1

Medium

Non Calculator

Let $f(x)=x(x-2)$ , for $0\le x\le 3$ .

A

Find the value of $\displaystyle \int_0^3 f(x)\,dx$ .

[2]

Write your answer here...

B

Find the total area between the graph of $f$ and the $x$ -axis for $0\le x\le 3$ .

[3]

Write your answer here...

0

Question 11

HL • Paper 1

Medium

Non Calculator

Consider the integral $I=\displaystyle \int_0^1 \sqrt{1-x^2}\,dx$ .

A

Using the substitution $x=\sin\theta$ , show that $I=\displaystyle \int_0^{\frac{\pi}{2}} \cos^2\theta\,d\theta$ .

[4]

Write your answer here...

B

Hence find the exact value of $I$ .

[3]

Write your answer here...

0

Question 12

HL • Paper 1

Medium

Non Calculator

A

Find the exact value of $\displaystyle \int_0^{\frac{\pi}{2}} x^2\cos x\,dx$ .

[5]

Write your answer here...

0

Question 13

HL • Paper 1

Medium

Non Calculator

The region $R$ is bounded by the curve $x=4-y^2$ and the $y$ -axis, for $0\le y\le 2$ .

A

Find the area of $R$ .

[2]

Write your answer here...

B

The region $R$ is rotated through $2\pi$ radians about the $y$ -axis. Find the volume of the solid formed.

[4]

Write your answer here...

0

Question 14

HL • Paper 1

Medium

Non Calculator

The curves $y=\sqrt{x}$ and $y=x$ enclose a region in the first quadrant.

A

Find the $x$ -coordinates of the points of intersection of the two curves.

[1]

Write your answer here...

B

The region is rotated through $2\pi$ radians about the $x$ -axis. Find the volume of the solid formed.

[4]

Write your answer here...

0

Question 15

SL • Paper 2

Medium

Calculator Permitted

Let

f(x)=x^3-4x^2+x+6

The graph of $f$ crosses the $x$ -axis at $x=-1$ , $x=2$ and $x=3$ .

A

Write down an integral expression for the total area enclosed by the graph of $f$ and the $x$ -axis between $x=-1$ and $x=3$ .

[2]

Write your answer here...

B

Find this total area.

[4]

Write your answer here...

0

Question 16

SL • Paper 2

Medium

Calculator Permitted

The functions $f$ and $g$ are defined by

f(x)=e^{0.4x}+1, \quad g(x)=0.3x^2+1.5

The graphs of $f$ and $g$ enclose two finite regions.

Exponential and quadratic curves with three crossings, forming two finite enclosed regions.

A

Find the $x$ -coordinates of all points of intersection of the two graphs.

[2]

Write your answer here...

B

Find the total area of the two finite regions enclosed by the graphs.

[4]

Write your answer here...

0

Question 17

HL • Paper 2

Medium

Calculator Permitted

Consider the integral

I=\int_0^1 \frac{x}{\sqrt{1+2x}}\,dx

Use the substitution $u=1+2x$ .

A

Show that

I=\frac14\int_1^3 \frac{u-1}{\sqrt{u}}\,du

[3]

Write your answer here...

B

Hence find the exact value of $I$ .

[3]

Write your answer here...

0

Question 18

HL • Paper 2

Medium

Calculator Permitted

Let

I=\int_0^2 x\ln(x+1)\,dx

A

Use integration by parts to find an exact expression for $I$ .

[5]

Write your answer here...

B

Give the value of $I$ to three significant figures.

[1]

Write your answer here...

0

Question 19

HL • Paper 2

Medium

Calculator Permitted

A curve is given by

x=y^2-4y, \quad 0\le y\le 4

The region $R$ is enclosed by the curve and the $y$ -axis.

A

Write down an integral expression for the area of $R$ .

[2]

Write your answer here...

B

Find the area of $R$ .

[2]

Write your answer here...

C

The region $R$ is rotated through $360^\circ$ about the $y$ -axis. Find the volume of the solid formed.

[2]

Write your answer here...

0

Question 20

SL • Paper 1

Medium

Non Calculator

A differentiable function $f$ is defined for $x>-\frac{1}{2}$ . It is given that

f'(x)=4(2x+1)^3-\frac{3}{2x+1}

and $f(0)=5$ .

A

I.

Find $f(x)$ in terms of $x$ and an arbitrary constant $C$ .

[3]

Write your answer here...

II.

Hence determine $f(x)$ .

[2]

Write your answer here...

B

I.

Find the exact value of $\displaystyle \int_0^1 f'(x)\,dx$ .

[3]

Write your answer here...

II.

Hence find $f(1)$ . If you did not obtain a value in part (b)(i), use $40-\frac{3}{2}\ln 3$ .

[2]

Write your answer here...

0

Question 21

SL • Paper 1

Medium

Non Calculator

For $x\ge 0$ , define

F(x)=\int_0^x 2t(t^2+4)^3\,dt

A

I.

Find $F(x)$ in terms of $x$ .

[3]

Write your answer here...

II.

Find $F(2)$ .

[2]

Write your answer here...

B

I.

Show that $F$ is an increasing function for $x\ge 0$ .

[2]

Write your answer here...

II.

Hence determine the value of $a$ , where $a\ge 0$ , such that $\displaystyle \int_0^a 2t(t^2+4)^3\,dt=960$ . If you did not obtain $F(2)$ in part (a)(ii), use $960$ .

[3]

Write your answer here...

0

Question 22

SL • Paper 1

Medium

Non Calculator

A function $f$ satisfies

f'(x)=6e^{3x}-2\cos(2x)

and $f(0)=1$ .

A

Part (a)

I.

Find $f(x)$ in terms of $x$ and an arbitrary constant $C$ .

[2]

Write your answer here...

II.

Hence determine $f(x)$ .

[2]

Write your answer here...

B

Find $f\left(\frac{\pi}{4}\right)$ .

[2]

Write your answer here...

C

Part (c)

I.

Find the exact value of $\displaystyle \int_0^{\frac{\pi}{4}} f'(x)\,dx$ . If you did not obtain $f\left(\frac{\pi}{4}\right)$ in part (b), use $2e^{\frac{3\pi}{4}}-2$ .

[2]

Write your answer here...

II.

Determine the exact value of $a$ such that $\displaystyle \int_0^a \left(f'(x)+2\cos(2x)\right)\,dx=14$ .

[3]

Write your answer here...

0

Question 23

HL • Paper 1

Medium

Non Calculator

Let $J=\displaystyle \int e^{2x}\sin x\,dx$ .

A

By using integration by parts twice, show that $J=\dfrac12e^{2x}\sin x-\dfrac14e^{2x}\cos x-\dfrac14J$ .

[4]

Write your answer here...

B

Hence find $\displaystyle \int e^{2x}\sin x\,dx$ .

[2]

Write your answer here...

0

Question 24

SL • Paper 2

Medium

Calculator Permitted

A small wind turbine produces power at a rate

P(t)=4+1.8\sin(0.4t)

kilowatts, where $t$ is the time in hours after 06:00 and $0\leq t\leq 12$ . The total energy produced, in kilowatt-hours, is found by integrating $P(t)$ with respect to $t$ .

Power output of the wind turbine over the 12-hour interval.

A

I.

Write down an integral expression for the total energy produced during the 12-hour period.

[1]

Write your answer here...

II.

Calculate this total energy.

[3]

Write your answer here...

B

I.

Find the average power produced during the 12-hour period.

[2]

Write your answer here...

II.

Explain why the energy produced from 06:00 to 18:00 is not equal to $12P(6)$ .

[1]

Write your answer here...

C

Determine the time after 06:00 at which half of the total energy has been produced.

[3]

Write your answer here...

0

Question 25

SL • Paper 2

Medium

Calculator Permitted

A curve $C$ has gradient function

\frac{dy}{dx}=3x^2e^{x^3-1}-2\cos(2x), \quad x\geq 0

The curve passes through the point $(0,5)$ .

A

I.

Find $y$ in terms of $x$ .

[4]

Write your answer here...

II.

Verify that your expression satisfies the given gradient function.

[1]

Write your answer here...

B

I.

Find the value of $y$ when $x=1.5$ .

[2]

Write your answer here...

II.

The point on $C$ with $x$ -coordinate $a$ has $y$ -coordinate $10$ , where $0<a<2$ . Find $a$ . If you did not obtain an expression for $y$ in part (a), use $y=e^{x^3-1}-\sin(2x)+4.63$ .

[3]

Write your answer here...

0

Question 26

SL • Paper 2

Medium

Calculator Permitted

The height, in metres, of an arch above a horizontal walkway is modelled by

h(x)=1+xe^{-0.25x^2}, \quad 0\leq x\leq 5

where $x$ is the horizontal distance in metres from one end of the arch.

Graph of the arch height function above the walkway.

A

I.

Find the area under the arch between $x=0$ and $x=5$ .

[3]

Write your answer here...

II.

Find the average height of the arch on this interval.

[1]

Write your answer here...

B

Show that the area under the arch from $x=0$ to $x=c$ is given by

c+2\left(1-e^{-0.25c^2}\right)

[3]

Write your answer here...

C

I.

Find the value of $c$ for which the area under the arch from $x=0$ to $x=c$ is $4\ \text{m}^2$ .

[2]

Write your answer here...

II.

State why there is only one such value of $c$ in the interval $0\leq c\leq 5$ .

[1]

Write your answer here...

0

Question 27

SL • Paper 2

Medium

Calculator Permitted

A curve has gradient function

\frac{dy}{dx}=2x\sqrt{x^2+5}, \quad x\ge 0

The curve passes through $(0,4)$ .

A

Find $y$ in terms of $x$ .

[4]

Write your answer here...

B

Find the value of $x$ for which $y=20$ .

[3]

Write your answer here...

0

Question 28

HL • Paper 2

Medium

Calculator Permitted

Consider the integral

I=\int_0^1 x^2e^{-x}\,dx

A

Use repeated integration by parts to find the exact value of $I$ .

[6]

Write your answer here...

B

Write down the value of $I$ to three significant figures.

[1]

Write your answer here...

0

Question 29

HL • Paper 2

Medium

Calculator Permitted

Let

I=\int e^{2x}\sin x\,dx

A

Use integration by parts twice to show that

I=\frac{e^{2x}}{5}(2\sin x-\cos x)+C

[5]

Write your answer here...

B

Hence find $\int_0^{\pi} e^{2x}\sin x\,dx$ to three significant figures.

[2]

Write your answer here...

0

Question 30

HL • Paper 2

Medium

Calculator Permitted

The region bounded by the curve

y=\ln(x+2)

the $x$ -axis and the line $x=4$ is rotated through $360^\circ$ about the $x$ -axis.

A

Find the $x$ -intercept of the curve.

[1]

Write your answer here...

B

Write down an integral expression for the volume of the solid formed.

[2]

Write your answer here...

C

Find this volume.

[4]

Write your answer here...

0

Question 31

HL • Paper 3

Medium

Calculator Permitted

A decorative lamp shade is modelled by rotating the curve $y=h\sin\left(\frac{\pi x}{L}\right),\quad 0\le x\le L$ through $2\pi$ radians about the $x$ -axis, where $h>0$ and $L>0$ .

Normalised lamp-shade arch about the x-axis.

A

I.

Show that $\int_0^L \sin^2\left(\frac{\pi x}{L}\right)\,dx=\frac{L}{2}$

[3]

Write your answer here...

II.

Hence show that the volume of the lamp shade is $V=\dfrac{\pi h^2L}{2}$ .

[2]

Write your answer here...

B

lamp shade has volume $500\text{ cm}^3$ and $L=10\text{ cm}$ . Find $h$ .

[2]

Write your answer here...

C

The value of $h$ from part (b) is kept fixed. Find the value of $L$ required to double the volume.

[1]

Write your answer here...

D

Find the area enclosed by the generating curve and the $x$ -axis in terms of $h$ and $L$ .

[2]

Write your answer here...

0

Question 32

SL • Paper 1

Hard

Non Calculator

The function $f$ is defined by

f(x)=(x+1)(x-2)(x-4), \quad -1\le x\le 4

A

Part (a)

I.

Write down the zeros of $f$ .

[1]

Write your answer here...

II.

Determine the sign of $f(x)$ on each of the intervals $-1<x<2$ and $2<x<4$ .

[2]

Write your answer here...

B

Part (b)

I.

Show that an antiderivative of $f$ is $F(x)=\frac{x^4}{4}-\frac{5x^3}{3}+x^2+8x$ .

[3]

Write your answer here...

II.

Find the value of $\displaystyle \int_{-1}^{4} f(x)\,dx$ . If you did not show the antiderivative in part (b)(i), use $F(x)=\frac{x^4}{4}-\frac{5x^3}{3}+x^2+8x$ .

[2]

Write your answer here...

C

Find the total area enclosed by the graph of $f$ and the $x$ -axis for $-1\le x\le 4$ .

[3]

Write your answer here...

0

Question 33

SL • Paper 1

Hard

Non Calculator

The functions $f$ and $g$ are defined by

f(x)=5-x^2, \qquad g(x)=x^2+1

The graphs of $f$ and $g$ enclose a finite region.

A

I.

Find the $x$ -coordinates of the points of intersection of the graphs.

[3]

Write your answer here...

II.

State which graph is above the other between the points of intersection.

[1]

Write your answer here...

B

Find the area of the finite region enclosed by the two graphs.

[4]

Write your answer here...

C

horizontal line $y=k$ , where $1<k<5$ , intersects the graph of $f$ at two points. The area enclosed by $y=f(x)$ and $y=k$ is $\frac{4}{3}$ . Find the value of $k$ .

[3]

Write your answer here...

0

Question 34

SL • Paper 1

Hard

Non Calculator

The function $f$ is defined by

f(x)=x-\frac{1}{x}, \quad \frac{1}{2}\le x\le 2

A

I.

Find the $x$ -intercept of the graph of $f$ on this interval.

[1]

Write your answer here...

II.

State the sign of $f(x)$ on each side of this intercept in the interval $\frac{1}{2}\le x\le 2$ .

[2]

Write your answer here...

B

I.

Find an antiderivative of $f(x)$ .

[2]

Write your answer here...

II.

Find the exact value of $\displaystyle \int_{\frac{1}{2}}^2 f(x)\,dx$ .

[2]

Write your answer here...

C

Find the total area between the graph of $f$ and the $x$ -axis on the interval $\frac{1}{2}\le x\le 2$ .

[3]

Write your answer here...

0

Question 35

HL • Paper 1

Hard

Non Calculator

Consider the definite integral

I=\int_0^1 \frac{x^3}{(1+x^2)^2}\,dx

A

I.

Use the substitution $u=1+x^2$ to show that

I=\frac{1}{2}\int_1^2 \frac{u-1}{u^2}\,du

[4]

Write your answer here...

II.

Hence find the exact value of $I$ .

[3]

Write your answer here...

B

The region under the curve $y=\dfrac{x^{\frac{3}{2}}}{1+x^2}$ , above the $x$ -axis, between $x=0$ and $x=1$ , is rotated through $2\pi$ radians about the $x$ -axis. Find the exact volume of the solid formed. If you did not obtain a value for $I$ , use $I=\frac{1}{2}\ln2-\frac{1}{4}$ .

[3]

Write your answer here...

0

Question 36

HL • Paper 1

Hard

Non Calculator

The curve $y=\ln x$ is considered for $1\le x\le e$ .

A

I.

Use integration by parts to show that

\int \ln x\,dx=x\ln x-x+C

[4]

Write your answer here...

II.

Hence find the area between the curve, the $x$ -axis, and the lines $x=1$ and $x=e$ .

[2]

Write your answer here...

B

I.

Use integration by parts and the result from part (a)(i) to show that

\int (\ln x)^2\,dx=x(\ln x)^2-2x\ln x+2x+C

[4]

Write your answer here...

II.

The region between $y=\ln x$ , the $x$ -axis, and the lines $x=1$ and $x=e$ is rotated through $2\pi$ radians about the $x$ -axis. Find the exact volume of the solid formed. If you did not show the result in part (b)(i), use it here.

[2]

Write your answer here...

0

Question 37

HL • Paper 1

Hard

Non Calculator

A curve is given by

x=e^y-1, \quad 0\le y\le \ln 3

The region $R$ is bounded by the curve, the $y$ -axis, and the lines $y=0$ and $y=\ln 3$ .

A

I.

Find the coordinates of the endpoints of the curve on the boundary of $R$ .

[2]

Write your answer here...

II.

Find the area of $R$ .

[3]

Write your answer here...

B

I.

The region $R$ is rotated through $2\pi$ radians about the $y$ -axis. Write down an integral expression for the volume of the solid formed.

[2]

Write your answer here...

II.

Hence find the exact volume of the solid formed.

[3]

Write your answer here...

0

Question 38

HL • Paper 1

Hard

Non Calculator

Consider the integral

I=\int_0^1 \frac{1}{1+\sqrt{x}}\,dx

A

I.

Use the substitution $u=\sqrt{x}$ to show that

I=\int_0^1 \frac{2u}{1+u}\,du

[4]

Write your answer here...

II.

Hence evaluate $I$ exactly.

[3]

Write your answer here...

B

The region under the curve $y=\dfrac{1}{\sqrt{1+\sqrt{x}}}$ , above the $x$ -axis, between $x=0$ and $x=1$ , is rotated through $2\pi$ radians about the $x$ -axis. Find the exact volume of the solid formed. If you did not obtain a value for $I$ , use $I=2-2\ln2$ .

[3]

Write your answer here...

0

Question 39

SL • Paper 2

Hard

Calculator Permitted

The functions $f$ and $g$ are defined by

f(x)=6e^{-0.3x}+1, \quad g(x)=0.8x+2, \quad x\geq 0

The region $R$ is bounded by the $y$ -axis and the graphs of $f$ and $g$ .

A

I.

Find the $x$ -coordinate of the point of intersection of the two graphs.

[2]

Write your answer here...

II.

State which curve is above the other for the region $R$ .

[1]

Write your answer here...

B

Find the area of $R$ .

[4]

Write your answer here...

C

I.

vertical line $x=k$ divides $R$ into two regions of equal area. Write down an equation that $k$ satisfies.

[2]

Write your answer here...

II.

Hence find $k$ . If you did not obtain the area in part (b), use $A=5.56$ .

[2]

Write your answer here...

0

Question 40

SL • Paper 2

Hard

Calculator Permitted

Let

f(x)=2\sin x-1, \quad 0\leq x\leq 2\pi

The graph of $f$ crosses the $x$ -axis twice on this interval.

A

I.

Find the two $x$ -intercepts of the graph of $f$ .

[2]

Write your answer here...

II.

Find the signed area $\displaystyle\int_0^{2\pi}f(x)\,\mathrm dx$ .

[2]

Write your answer here...

B

Find the total area between the graph of $f$ and the $x$ -axis for $0\leq x\leq 2\pi$ .

[4]

Write your answer here...

C

I.

Let $k$ be the value such that the total area between the graph and the $x$ -axis from $x=0$ to $x=k$ is half of the total area found in part (b). Write down an equation for $k$ .

[2]

Write your answer here...

II.

Hence find $k$ . If you did not obtain the total area in part (b), use $A=9.02$ .

[2]

Write your answer here...

0

Question 41

HL • Paper 2

Hard

Calculator Permitted

The curve $C$ has equation

y=xe^{-0.5x}, \quad 0\leq x\leq 5

The region $R$ is bounded by $C$ , the $x$ -axis, and the lines $x=0$ and $x=5$ .

A

I.

Use integration by parts to find the area of $R$ .

[4]

Write your answer here...

II.

Find the maximum height of the region $R$ .

[1]

Write your answer here...

B

The region $R$ is rotated through $2\pi$ radians about the $x$ -axis. Show that the volume of the solid formed is

V=\pi\int_0^5x^2e^{-x}\,\mathrm dx

[2]

Write your answer here...

C

I.

Use repeated integration by parts to find $\displaystyle\int_0^5x^2e^{-x}\,\mathrm dx$ exactly.

[4]

Write your answer here...

II.

Hence find the volume of the solid formed.

[2]

Write your answer here...

0

Question 42

HL • Paper 2

Hard

Calculator Permitted

A plane region $R$ lies between the curve

x=y\sqrt{9-y^2}, \quad 0\leq y\leq 3

and the $y$ -axis.

A

I.

Write down an integral expression for the area of $R$ .

[1]

Write your answer here...

II.

Use the substitution $u=9-y^2$ to find the area of $R$ .

[4]

Write your answer here...

B

The region $R$ is rotated through $2\pi$ radians about the $y$ -axis. Find the volume of the solid formed.

[4]

Write your answer here...

C

I.

Let $y=k$ divide the solid into two parts of equal volume. Write down an equation for $k$ .

[1]

Write your answer here...

II.

Hence find $k$ . If you did not obtain the volume in part (b), use $V=32.4\pi$ .

[2]

Write your answer here...

0

Question 43

HL • Paper 2

Hard

Calculator Permitted

The region $R$ is bounded by the curve

y=\ln(2x+1)

the $x$ -axis, the $y$ -axis, and the line $x=4$ .

Curve with shaded region R and the boundary segment x = 4.

A

I.

Use integration by parts to find the exact area of $R$ .

[3]

Write your answer here...

II.

Write this area correct to three significant figures.

[1]

Write your answer here...

B

The region $R$ is rotated through $2\pi$ radians about the $x$ -axis. Write down an integral expression for the volume $V$ of the solid formed, and calculate $V$ .

[4]

Write your answer here...

C

I.

Let $x=a$ divide the solid into two parts of equal volume. Write down an equation for $a$ .

[2]

Write your answer here...

II.

Hence find $a$ . If you did not obtain the volume in part (b), use $V=31.3$ .

[2]

Write your answer here...

0

Question 44

HL • Paper 3

Hard

Calculator Permitted

For $a>0$ , define $I(a)=\int_0^1 \frac{2x}{1+a x^2}\,dx$ This question investigates how $I(a)$ varies with the parameter $a$ .

Decreasing curve of I(a)=ln(1+a)/a with a horizontal line at I(a)=0.5.

A

I.

Use the substitution $u=1+a x^2$ to express $I(a)$ as an integral with respect to $u$ .

[2]

Write your answer here...

II.

Hence show that $I(a)=\dfrac{\ln(1+a)}{a}$ .

[2]

Write your answer here...

B

Find the value of $a$ for which $I(a)=0.5$ .

[2]

Write your answer here...

C

Prove that $I(a)$ is a decreasing function of $a$ for $a>0$ .

[3]

Write your answer here...

D

Deduce the number of positive solutions of $I(a)=m$ when $0<m<1$ .

[2]

Write your answer here...

0

Question 45

HL • Paper 3

Hard

Calculator Permitted

For $n\in\mathbb{Z}_{\ge 0}$ , let

I_n=\int_0^1 x^n e^x\,dx

This question investigates a recurrence relation for $I_n$ .

$n$	$I_n$
4	$9e-24\approx0.4645$
5	$120-44e\approx0.3956$
6	$265e-720\approx0.3447$
7	$5040-1854e\approx0.3055$
8	$14833e-40320\approx0.2744$
9	$362880-133497e\approx0.2490$
10	$1334971e-3628800\approx0.2280$
11	$39916800-14684681e\approx0.2103$

A

I.

Write down the exact value of $I_0$ .

[1]

Write your answer here...

II.

Use integration by parts to show that $I_n=e-nI_{n-1}$ for $n\ge 1$ .

[4]

Write your answer here...

B

Hence find the exact values of $I_1$ , $I_2$ and $I_3$ .

[3]

Write your answer here...

C

Show that $0<I_n<\dfrac{e}{n+1}$ for $n\ge 0$ .

[2]

Write your answer here...

D

Use the result in part (c) to find an integer $N$ such that $I_n<0.01$ for all $n\ge N$ .

[2]

Write your answer here...

0

Question 46

HL • Paper 3

Hard

Calculator Permitted

For $0<k<1$ , the curve $y=e^{-x}$ and the line $y=k$ enclose a finite region with the $y$ -axis. The region is denoted by $R_k$ .

Illustrative graph for k₀=0.6 showing y=e^{-x}, y=k₀, and the shaded region R_{k₀} bounded by the y-axis.

A

I.

Find the $x$ -coordinate of the point where $y=e^{-x}$ meets $y=k$ .

[1]

Write your answer here...

II.

Show that the area of $R_k$ is $A(k)=1-k+k\ln k$ .

[3]

Write your answer here...

B

Find $A'(k)$ and interpret its sign for $0<k<1$ .

[2]

Write your answer here...

C

Find the value of $k$ for which the area of $R_k$ is $0.25$ .

[2]

Write your answer here...

D

For this value of $k$ , find the volume generated when $R_k$ is rotated through $2\pi$ radians about the $x$ -axis. If you did not obtain a value for $k$ , use $k=0.383$ .

[3]

Write your answer here...

0

Question 47

HL • Paper 3

Hard

Calculator Permitted

A region $R$ is bounded by the curve $x=y^2(3-y),\quad 0\le y\le 3$ and the $y$ -axis. The curve lies to the right of the $y$ -axis on this interval.

A

Find the area of $R$ .

[3]

Write your answer here...

B

Find the volume generated when $R$ is rotated through $2\pi$ radians about the $y$ -axis.

[3]

Write your answer here...

C

I.

horizontal line $y=c$ divides $R$ into two regions of equal area. Write an equation satisfied by $c$ .

[1]

Write your answer here...

II.

Find $c$ .

[2]

Write your answer here...

D

Explain why integration with respect to $y$ is the natural method for this region.

[1]

Write your answer here...

0

Question 48

HL • Paper 3

Hard

Calculator Permitted

The region $R$ is bounded by the curve $y=-\ln x$ , the $x$ -axis, and the vertical lines $x=e^{-1}$ and $x=1$ .

Shaded region R under $y=-\ln x$, bounded by $x=e^{-1}$ and $x=1$.

A

Find the exact area of $R$ .

[3]

Write your answer here...

B

I.

Use integration by parts to show that

\int (\ln x)^2\,dx=x\left((\ln x)^2-2\ln x+2\right)+C

[3]

Write your answer here...

C

Find the exact volume generated when $R$ is rotated through $2\pi$ radians about the $x$ -axis.

[2]

Write your answer here...

D

vertical line $x=k$ divides the volume in part (c) into two equal volumes. Find $k$ .

[3]

Write your answer here...

0

Question 49

HL • Paper 3

Hard

Calculator Permitted

The curve

y=x\sqrt{4-x^2},\quad 0\le x\le 2

forms a single arch above the $x$ -axis. The region under the arch is denoted by $R$ .

A

I.

Use the substitution $u=4-x^2$ to find the exact area of $R$ .

[3]

Write your answer here...

II.

State why the definite integral gives the geometric area without splitting the interval.

[1]

Write your answer here...

B

Find the volume generated when $R$ is rotated through $2\pi$ radians about the $x$ -axis.

[3]

Write your answer here...

C

vertical plane perpendicular to the $x$ -axis cuts the solid into two parts of equal volume at $x=c$ . Find $c$ .

[3]

Write your answer here...

0

Question 50

HL • Paper 3

Hard

Calculator Permitted

A region $R_k$ is bounded by the curve

x=\ln(y+1)

the $y$ -axis, and the horizontal lines $y=0$ and $y=k$ , where $k>0$ .

Boundary curve x=ln(y+1) with the coordinate axes.

A

Write down an integral expression for the area of $R_k$ .

[1]

Write your answer here...

B

I.

Use the substitution $u=y+1$ and integration by parts to show that

\int \ln^2(y+1)\,dy=(y+1)(\ln^2(y+1)-2\ln(y+1)+2)+C

[3]

Write your answer here...

II.

Write down an integral expression for the volume formed when $R_k$ is rotated through $2\pi$ radians about the $y$ -axis.

[1]

Write your answer here...

C

Show that $V=\pi\left((k+1)(\ln^2(k+1)-2\ln(k+1)+2)-2\right)$

[2]

Write your answer here...

D

Find the value of $k$ for which $V=10$ .

[3]

Write your answer here...

0

Question 51

HL • Paper 3

Hard

Calculator Permitted

A density profile along a beam is modelled by $f(x)=x e^{-\frac{x^2}{2}},\quad x\ge 0$ For $a>0$ , define $A(a)=\int_0^a x e^{-\frac{x^2}{2}}\,dx$

A

I.

Use the reverse chain rule to show that $A(a)=1-e^{-\frac{a^2}{2}}$ .

[3]

Write your answer here...

II.

Find $\lim\limits_{a\to\infty}A(a)$ .

[1]

Write your answer here...

B

Find $a$ such that $A(a)=0.8$ .

[2]

Write your answer here...

C

The region under the curve from $x=0$ to the value of $a$ found in part (b) is rotated through $2\pi$ radians about the $x$ -axis. Write down an integral expression for the volume and find its value. If you did not obtain $a$ , use $a=1.79$ .

[4]

Write your answer here...

0

Question 52

HL • Paper 1

Hard

Non Calculator

Let

I=\int_0^1 x^2e^{2x}\,dx

A

I.

By using integration by parts, show that

I=\frac{e^2}{2}-\int_0^1 xe^{2x}\,dx

[3]

Write your answer here...

II.

Hence evaluate $I$ exactly.

[4]

Write your answer here...

B

I.

Find the exact value of $\displaystyle \int_0^1 x(1-x)e^{2x}\,dx$ . If you did not obtain $I$ in part (a)(ii), use $I=\frac{e^2-1}{4}$ .

[2]

Write your answer here...

II.

Determine the exact value of $k$ such that

\int_0^1 (x^2-k)e^{2x}\,dx=0

[3]

Write your answer here...

0

Question 53

HL • Paper 1

Hard

Non Calculator

Let

I=\int_0^{\frac{\pi}{2}}e^x\sin x\,dx, \qquad J=\int_0^{\frac{\pi}{2}}e^x\cos x\,dx

A

I.

Show that $I=e^{\frac{\pi}{2}}-J$ .

[2]

Write your answer here...

II.

Show that $J=I-1$ .

[2]

Write your answer here...

B

Hence find the exact values of $I$ and $J$ .

[3]

Write your answer here...

C

Find the total area between the graph of $y=e^x(\sin x-\cos x)$ and the $x$ -axis for $0\le x\le \frac{\pi}{2}$ .

[4]

Write your answer here...

0

Question 54

SL • Paper 2

Hard

Calculator Permitted

Water enters a tank at a rate

R(t)=\frac{20}{1+0.15t^2}

litres per minute, where $t$ is the time in minutes after the tap is opened. At the same time, water leaves the tank at a constant rate of $4$ litres per minute. Initially the tank contains $60$ litres of water.

Water inflow and outflow rates over the first 8 minutes.

A

I.

Write down an integral expression for the net change in the volume of water in the tank during the first 8 minutes.

[1]

Write your answer here...

II.

Calculate this net change.

[3]

Write your answer here...

B

I.

Find the time at which the accumulated volume predicted by the inflow and outflow rates is greatest during the first 8 minutes, ignoring any capacity limit.

[2]

Write your answer here...

II.

Find this greatest accumulated volume, ignoring any capacity limit.

[2]

Write your answer here...

C

The capacity of the tank is $90$ litres. Determine the time at which the tank first overflows.

[3]

Write your answer here...

D

Explain why the tank is not still overflowing at $t=8$ .

[2]

Write your answer here...

0

Question 55

HL • Paper 2

Hard

Calculator Permitted

Let

f(x)=e^{-x}\sin(2x), \quad 0\leq x\leq \pi

The function is positive on $0<x<\dfrac{\pi}{2}$ and negative on $\dfrac{\pi}{2}<x<\pi$ .

Graph of f(x)=e^{-x}sin(2x) on 0≤x≤π, with zeros marked.

A

I.

Use integration by parts twice to show that

\int e^{-x}\sin(2x)\,\mathrm dx=\frac{e^{-x}(-\sin(2x)-2\cos(2x))}{5}+C

[4]

Write your answer here...

II.

Hence find the signed area $\displaystyle\int_0^\pi f(x)\,\mathrm dx$ .

[1]

Write your answer here...

B

Find the total area between the graph of $f$ and the $x$ -axis for $0\leq x\leq \pi$ .

[3]

Write your answer here...

C

I.

The value $k$ , where $0<k<\dfrac{\pi}{2}$ , is such that the area between the graph and the $x$ -axis from $0$ to $k$ is half of the positive lobe area. Write down an equation for $k$ .

[2]

Write your answer here...

II.

Hence find $k$ . If you did not obtain the positive lobe area, use $0.483$ .

[2]

Write your answer here...

0

Question 56

HL • Paper 2

Hard

Calculator Permitted

The region $R$ is bounded by the curve

y=\frac{\sqrt{x}}{1+x}, \quad 0\leq x\leq 4

the $x$ -axis, and the lines $x=0$ and $x=4$ .

Curve and bounded region under $y=\frac{\sqrt{x}}{1+x}$ for $0\le x\le 4$, with the x-axis and boundary $x=4$ shown.

A

I.

Use the substitution $u=\sqrt{x}$ to show that the area of $R$ is

4-2\arctan 2

[4]

Write your answer here...

II.

Find the area of $R$ to three significant figures.

[1]

Write your answer here...

B

The region $R$ is rotated through $2\pi$ radians about the $x$ -axis. Find the volume of the solid formed.

[4]

Write your answer here...

C

I.

Let $x=c$ divide the solid into two parts of equal volume. Write down an equation for $c$ .

[1]

Write your answer here...

II.

Hence find $c$ , taking the root in the interval $0<c<4$ . If you did not obtain the volume in part (b), use $V=2.54$ .

[2]

Write your answer here...

0

Question 57

HL • Paper 3

Hard

Calculator Permitted

For real $a>0$ , define $K(a)=\int_0^{\frac{\pi}{2}} e^{ax}\cos x\,dx$ This question first considers the case $a=2$ and then generalizes.

Family of curves for y=e^{ax}cos x on 0≤x≤π/2.

A

I.

Using integration by parts twice, show that $\int e^{2x}\cos x\,dx=\frac{e^{2x}(2\cos x+\sin x)}{5}+C$

[4]

Write your answer here...

II.

Hence find $K(2)$ exactly.

[1]

Write your answer here...

B

By considering $\int e^{ax}\cos x\,dx$ , show that $K(a)=\frac{e^{\frac{a\pi}{2}}-a}{a^2+1}$

[4]

Write your answer here...

C

Find the positive value of $a$ for which $K(a)=10$ .

[2]

Write your answer here...

D

Explain why no value $0<a<2$ can satisfy $K(a)=10$ .

[1]

Write your answer here...

0

Question 58

HL • Paper 3

Hard

Calculator Permitted

The tangent half-angle substitution is $t=\tan\frac{x}{2},\quad \cos x=\frac{1-t^2}{1+t^2},\quad dx=\frac{2}{1+t^2}\,dt$ . This question investigates $I(a)=\int_0^{\frac{\pi}{2}}\frac{1}{a+\cos x}\,dx,\quad a>1$ .

A

I.

For $a=2$ , show that the integral becomes $\int_0^1 \frac{2}{3+t^2}\,dt$ .

[2]

Write your answer here...

II.

Hence find $I(2)$ exactly.

[2]

Write your answer here...

B

Show that, for $a>1$ ,

I(a)=\frac{2}{\sqrt{a^2-1}}\arctan\sqrt{\frac{a-1}{a+1}}

[4]

Write your answer here...

C

I.

Find $\lim\limits_{a\to 1^+} I(a)$ directly from the original integral.

[2]

Write your answer here...

II.

State what this suggests about the formula in part (b) as $a\to 1^+$ .

[1]

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0

Question 59

HL • Paper 3

Hard

Calculator Permitted

For $n\in\mathbb{Z}_{\ge 0}$ , define

A_n=\int_0^1 x^n\ln(1+x)\,dx,\qquad B_n=\int_0^1\frac{x^n}{1+x}\,dx

n	A_n	nA_n
1	0.250000	0.250000
2	0.184320	0.368641
5	0.102778	0.513889
10	0.059068	0.590682
20	0.031900	0.638004
50	0.013401	0.670039

A

Find the exact value of $A_0$ .

[2]

Write your answer here...

B

I.

Use integration by parts to show that $A_n=\frac{\ln2}{n+1}-\frac{B_{n+1}}{n+1}$

[3]

Write your answer here...

II.

Show that $B_n+B_{n-1}=\dfrac1n$ for $n\ge 1$ .

[1]

Write your answer here...

C

Find the exact values of $A_1$ and $A_2$ .

[4]

Write your answer here...

D

Using the table or technology, estimate $\lim\limits_{n\to\infty} nA_n$ and explain why this value is reasonable.

[2]

Write your answer here...

0

Question 60

HL • Paper 2

Hard

Calculator Permitted

For $k>0$ , a family of profile curves is given by

x=ye^{-ky}, \quad 0\leq y\leq 4

The region $R_k$ is bounded by the curve, the $y$ -axis, and the lines $y=0$ and $y=4$ .

Representative curve x = y e^(-0.25y) with the boundary y = 4.

A

For this part, take $k=0.25$ .

I.

Write down an integral expression for the area of $R_{0.25}$ .

[1]

Write your answer here...

II.

Use integration by parts to find this area.

[4]

Write your answer here...

B

I.

The region $R_{0.25}$ is rotated through $2\pi$ radians about the $y$ -axis. Write down an integral expression for the volume $V$ of the solid formed.

[2]

Write your answer here...

II.

Calculate $V$ .

[3]

Write your answer here...

C

I.

Let $y=h$ divide the solid in part (b) into two parts of equal volume. Write down an equation for $h$ .

[1]

Write your answer here...

II.

Hence find $h$ . If you did not obtain the volume in part (b), use $V=16.3$ .

[2]

Write your answer here...

D

For general $k>0$ , let

V(k)=\pi\int_0^4 y^2e^{-2ky}\,\mathrm dy

Justify that $V(k)$ decreases as $k$ increases, and determine the value of $k$ for which $V(k)=10$ .

[2]

Write your answer here...

0