Integral Calculus - IB Math AA Question Bank

Subjects

Features

Review

HOT

Tutoring

Integral Calculus

Practice exam-style IB Math AA questions for Integral Calculus, aligned with the syllabus and grouped by topic.

Verified by Karim

Paper

Difficulty

Status

Level

This content can't be printed from the browser. Use Export PDF with Pro for an offline copy.

Question 1

SL • Paper 2

Easy

Calculator Permitted

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 $(4,18)$ .

Find $y$ in terms of $x$ .

[4]

Question 2

SL • Paper 1

Medium

Non Calculator

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 $f(1)=4$ .

Find $f(x)$ .

[5]

Question 3

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

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

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

[2]

Find the area of the enclosed region.

[3]

Question 4

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

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

[4]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 5

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

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

Find $f'(x)$ .

[3]

Hence find $f(x)$ .

[2]

Question 6

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

[4]

Question 7

SL • Paper 2

Medium

Calculator Permitted

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 $f(1)=5$ .

Find an expression for $f(x)$ .

[4]

Find $f(3)$ .

[2]

Question 8

SL • Paper 2

Medium

Calculator Permitted

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.

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

[2]

Find the area of the shaded region.

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 9

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

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

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

[1]

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

[5]

Question 10

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

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

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

[2]

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

[3]

Question 11

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

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

[4]

Hence find the exact value of $I$ .

[3]

Question 12

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

[5]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 13

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

Find the area of $R$ .

[2]

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

[4]

Question 14

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

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

[1]

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

[4]

Question 15

SL • Paper 2

Medium

Calculator Permitted

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 $x=3$ .

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]

Find this total area.

[4]

Question 16

SL • Paper 2

Medium

Calculator Permitted

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.

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

[2]

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

[4]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 17

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the integral

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

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

Show that

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

[3]

Hence find the exact value of $I$ .

[3]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

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

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

[5]

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

[1]

Question 19

HL • Paper 2

Medium

Calculator Permitted

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.

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

[2]

Find the area of $R$ .

[2]

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

[2]

Question 20

SL • Paper 1

Medium

Non Calculator

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 $f(0)=5$ .

I.

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

[3]

II.

Hence determine $f(x)$ .

[2]

I.

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

[3]

II.

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 21

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

For $x\ge 0$ , define

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

I.

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

[3]

II.

Find $F(2)$ .

[2]

I.

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

[2]

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]

Question 22

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A function $f$ satisfies

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

and $f(0)=1$ .

Part (a)

I.

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

[2]

II.

Hence determine $f(x)$ .

[2]

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

[2]

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]

II.

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

[3]

Question 23

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

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

[4]

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

[2]

Question 24

SL • Paper 2

Medium

Calculator Permitted

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 $t$ .

I.

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

[1]

II.

Calculate this total energy.

[3]

I.

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

[2]

II.

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

[1]

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

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 25

SL • Paper 2

Medium

Calculator Permitted

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 $(0,5)$ .

I.

Find $y$ in terms of $x$ .

[4]

II.

Verify that your expression satisfies the given gradient function.

[1]

I.

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

[2]

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]

Question 26

SL • Paper 2

Medium

Calculator Permitted

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.

I.

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

[3]

II.

Find the average height of the arch on this interval.

[1]

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]

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]

II.

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

[1]

Question 27

SL • Paper 2

Medium

Calculator Permitted

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 $(0,4)$ .

Find $y$ in terms of $x$ .

[4]

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

[3]

Question 28

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the integral

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

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

[6]

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

[1]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 29

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

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

Use integration by parts twice to show that

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

[5]

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

[2]

Question 30

HL • Paper 2

Medium

Calculator Permitted

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.

Find the $x$ -intercept of the curve.

[1]

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

[2]

Find this volume.

[4]

Question 31

HL • Paper 3

Medium

Calculator Permitted

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 $L>0$ .

I.

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

[3]

II.

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

[2]

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

[2]

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

[1]

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

[2]

Question 32

SL • Paper 1

Hard

Non Calculator

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

Part (a)

I.

Write down the zeros of $f$ .

[1]

II.

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

[2]

Part (b)

I.

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

[3]

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]

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

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 33

SL • Paper 1

Hard

Non Calculator

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.

I.

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

[3]

II.

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

[1]

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

[4]

A 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]

Question 34

SL • Paper 1

Hard

Non Calculator

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

I.

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

[1]

II.

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

[2]

I.

Find an antiderivative of $f(x)$ .

[2]

II.

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

[2]

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

[3]

Question 35

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the definite integral

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

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]

II.

Hence find the exact value of $I$ .

[3]

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]

Question 36

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

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

I.

Use integration by parts to show that

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

[4]

II.

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

[2]

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]

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]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 37

HL • Paper 1

Hard

Non Calculator

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 $y=\ln 3$ .

I.

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

[2]

II.

Find the area of $R$ .

[3]

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]

II.

Hence find the exact volume of the solid formed.

[3]

Question 38

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the integral

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

I.

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

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

[4]

II.

Hence evaluate $I$ exactly.

[3]

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]

Question 39

SL • Paper 2

Hard

Calculator Permitted

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 $g$ .

I.

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

[2]

II.

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

[1]

Find the area of $R$ .

[4]

I.

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

[2]

II.

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

[2]

Question 40

SL • Paper 2

Hard

Calculator Permitted

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.

I.

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

[2]

II.

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

[2]

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

[4]

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]

II.

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 41

HL • Paper 2

Hard

Calculator Permitted

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 $x=5$ .

I.

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

[4]

II.

Find the maximum height of the region $R$ .

[1]

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]

I.

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

[4]

II.

Hence find the volume of the solid formed.

[2]

Question 42

HL • Paper 2

Hard

Calculator Permitted

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.

I.

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

[1]

II.

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

[4]

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

[4]

I.

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

[1]

II.

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

[2]

Question 43

HL • Paper 2

Hard

Calculator Permitted

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 $x=4$ .

I.

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

[3]

II.

Write this area correct to three significant figures.

[1]

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]

I.

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

[2]

II.

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

[2]

Question 44

HL • Paper 3

Hard

Calculator Permitted

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 $a$ .

I.

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

[2]

II.

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

[2]

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

[2]

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

[3]

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

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 45

HL • Paper 3

Hard

Calculator Permitted

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	$39916800-14684681e\approx0.2103$

I.

Write down the exact value of $I_0$ .

[1]

II.

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

[4]

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

[3]

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

[2]

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

[2]

Question 46

HL • Paper 3

Hard

Calculator Permitted

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 $R_k$ .

I.

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

[1]

II.

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

[3]

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

[2]

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

[2]

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]

Question 47

HL • Paper 3

Hard

Calculator Permitted

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.

Find the area of $R$ .

[3]

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

[3]

I.

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

[1]

II.

Find $c$ .

[2]

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

[1]

Question 48

HL • Paper 3

Hard

Calculator Permitted

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 $x=1$ .

Find the exact area of $R$ .

[3]

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]

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

[2]

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

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 49

HL • Paper 3

Hard

Calculator Permitted

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 $R$ .

I.

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

[3]

II.

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

[1]

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

[3]

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

[3]

Question 50

HL • Paper 3

Hard

Calculator Permitted

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 $k>0$ .

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

[1]

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]

II.

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

[1]

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

[2]

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

[3]

Question 51

HL • Paper 3

Hard

Calculator Permitted

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 $A(a)=\int_0^a x e^{-\frac{x^2}{2}}\,dx$

I.

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

[3]

II.

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

[1]

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

[2]

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]

Question 52

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let

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

I.

By using integration by parts, show that

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

[3]

II.

Hence evaluate $I$ exactly.

[4]

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]

II.

Determine the exact value of $k$ such that

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

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 53

HL • Paper 1

Hard

Non Calculator

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

I.

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

[2]

II.

Show that $J=I-1$ .

[2]

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

[3]

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]

Question 54

SL • Paper 2

Hard

Calculator Permitted

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.

I.

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

[1]

II.

Calculate this net change.

[3]

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]

II.

Find this greatest accumulated volume, ignoring any capacity limit.

[2]

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

[3]

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

[2]

Question 55

HL • Paper 2

Hard

Calculator Permitted

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 $\dfrac{\pi}{2}<x<\pi$ .

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]

II.

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

[1]

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

[3]

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]

II.

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

[2]

Question 56

HL • Paper 2

Hard

Calculator Permitted

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 $x=4$ .

I.

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

4-2\arctan 2

[4]

II.

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

[1]

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

[4]

I.

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

[1]

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]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 57

HL • Paper 3

Hard

Calculator Permitted

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.

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]

II.

Hence find $K(2)$ exactly.

[1]

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

[4]

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

[2]

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

[1]

Question 58

HL • Paper 3

Hard

Calculator Permitted

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 $I(a)=\int_0^{\frac{\pi}{2}}\frac{1}{a+\cos x}\,dx,\quad a>1$ .

I.

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

[2]

II.

Hence find $I(2)$ exactly.

[2]

Show that, for $a>1$ ,

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

[4]

I.

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

[2]

II.

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

[1]

Question 59

HL • Paper 3

Hard

Calculator Permitted

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

Find the exact value of $A_0$ .

[2]

I.

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

[3]

II.

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

[1]

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

[4]

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

[2]

Question 60

HL • Paper 2

Hard

Calculator Permitted

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 $y=4$ .

For this part, take $k=0.25$ .

I.

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

[1]

II.

Use integration by parts to find this area.

[4]

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]

II.

Calculate $V$ .

[3]

I.

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

[1]

II.

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

[2]

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]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Differential Equations

Kinematics

The Ultimate All-in-One
IB Platform

All content on this website has been developed independently from and is not endorsed by the International Baccalaureate Organization. International Baccalaureate and IB are registered trademarks owned by the International Baccalaureate Organization.