Integration - IB Math AI Question Bank

Question 1

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

A function $F$ satisfies

F'(x)=6x^2-8x+5

and $F(1)=7$ .

A

Find the general form of $F(x)$ .

[2]

B

Determine the value of $C$ .

[1]

C

Hence find $F(3)$ .

[2]

Question 2

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

The cross-section of a drainage channel is modelled by the region between the curve $y=-x^2+6x$ and the $x$ -axis, for $0\leq x\leq6$ . Both $x$ and $y$ are measured in metres.

A

Write down an integral that represents the cross-sectional area of the channel.

[1]

B

Find the cross-sectional area of the channel.

[2]

C

Find the mean depth of the channel across its width.

[2]

Question 3

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

A flower bed is modelled by the region between the curve

y=(x-1)(5-x)

and the $x$ -axis. Both coordinates are measured in metres.

A

Find the two $x$ -intercepts of the curve.

[1]

B

Write down an integral representing the area of the flower bed.

[1]

C

Find the area of the flower bed.

[2]

Question 4

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

The gradient of a curve is given by

\frac{\mathrm{d}y}{\mathrm{d}x}=9x^2+4

The curve passes through the point $(-1,2)$ .

A

Find an expression for $y$ in terms of $x$ and an arbitrary constant $C$ .

[2]

B

Determine the value of $C$ .

[1]

C

Find the value of $y$ when $x=2$ .

[1]

Question 5

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

Water enters a tank at a rate of

r(t)=0.6t^2-2.4t+5

litres per minute, where $t$ is the time in minutes after a pump is switched on. Initially, the tank contains $12$ litres of water.

A

Write down an integral representing the volume of water that enters the tank during the first $6$ minutes.

[1]

B

Calculate the volume of water that enters during the first $6$ minutes.

[2]

C

Find the volume of water in the tank after $6$ minutes.

[1]

D

Calculate the average rate at which water enters the tank during these $6$ minutes.

[2]

Question 6

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

The velocity of a cyclist was recorded every $2$ seconds from $t=0$ to $t=8$ seconds. The results are shown in the table.

time [s]	velocity [m s^-1]
0	3.2
2	4.1
4	5.0
6	4.6
8	3.5

A

The recorded velocities, in order, are $3.2$ , $4.1$ , $5.0$ , $4.6$ and $3.5\ \text{m s}^{-1}$ . Use the trapezoidal rule to estimate the distance travelled during the $8$ seconds.

[3]

B

A tracking device recorded the actual distance as $33.4\ \text{m}$ . Calculate the percentage error in the trapezoidal estimate, stating whether it is an overestimate or an underestimate.

[2]

Question 7

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

The area under the curve $y=x^2+1$ , from $x=0$ to $x=2$ , is to be approximated using four intervals of equal width.

A

Use the trapezoidal rule with ordinates at $x=0$ , $0.5$ , $1$ , $1.5$ and $2$ to estimate the area.

[2]

B

Use integration to find the exact area.

[2]

C

Explain why the trapezoidal rule gives an overestimate in this case.

[1]

Question 8

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A function $F$ is defined for $x>0$ and satisfies

F'(x)=3\sqrt{x}-\frac{4}{x}

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

A

Find the general form of $F(x)$ .

[2]

B

Determine the value of $C$ .

[1]

C

Hence find $F(4)$ .

[2]

Question 9

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The rate of change of a quantity $Q$ is modelled by

\frac{\mathrm{d}Q}{\mathrm{d}t}=6\cos(3t)

for $0\leq t\leq\dfrac{\pi}{6}$ , where $t$ is measured in seconds.

A

Find an antiderivative of $6\cos(3t)$ .

[2]

B

Find the exact change in $Q$ over the given interval.

[1]

C

Find the average value of $\dfrac{\mathrm{d}Q}{\mathrm{d}t}$ over the interval, giving your answer exactly.

[2]

Question 10

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The region bounded by the curve $y=\sqrt{4-x}$ , the $x$ -axis and the $y$ -axis is rotated through $2\pi$ radians about the $x$ -axis.

A first-quadrant coordinate diagram showing a decreasing curve from the positive y-axis to the positive x-axis. The region under the curve and between the coordinate axes is shaded, and the x-axis is identified as the axis of rotation.

A

Write down an integral representing the volume of the solid formed.

[2]

B

Find the exact volume of the solid and give its value to three significant figures.

[3]

Question 11

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

A surveyor measures the width of a lake at $25$ metre intervals along a straight baseline of length $150$ metres. The measurements are shown in the table.

Distance along baseline [m]	Lake width [m]
0	0
25	18
50	27
75	31
100	24
125	15
150	0

A

The measured widths, in order, are $0$ , $18$ , $27$ , $31$ , $24$ , $15$ and $0$ metres. Use the trapezoidal rule to estimate the surface area of the lake.

[4]

B

A satellite estimate gives the surface area as $3010\ \text{m}^2$ . Calculate the percentage by which the trapezoidal estimate is less than the satellite estimate.

[2]

Question 12

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The mass of a substance in a container increases at a rate

r(t)=2t e^{t^2+1}

grams per hour, where $0\leq t\leq1$ . Initially, the container holds $10$ grams of the substance.

A

Using the substitution $u=t^2+1$ , find an antiderivative of $r(t)$ .

[2]

B

Find the exact increase in mass during the first hour.

[1]

C

Calculate the mass in the container after one hour.

[2]

Question 13

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

For $x\geq0$ , a rate is modelled by

r(x)=\frac{12}{3x+2}

A

Find $\displaystyle\int r(x)\,\mathrm{d}x$ .

[2]

B

Show that the accumulated amount from $x=0$ to $x=a$ is

4\ln\left(\frac{3a+2}{2}\right)

[2]

C

Hence determine $a$ if the accumulated amount is $4\ln5$ .

[2]

Question 14

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The curve $y=(x-1)(x-4)$ is considered on the interval $0\leq x\leq5$ .

Parabola $y=(x-1)(x-4)$ on $0\leq x\leq5$.

A

Find the value of $\displaystyle\int_0^5(x-1)(x-4)\,\mathrm{d}x$ .

[2]

B

Explain why the answer to part (a) is not the total geometric area between the curve and the $x$ -axis.

[1]

C

Find the total geometric area between the curve and the $x$ -axis for $0\leq x\leq5$ .

[3]

Question 15

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A region is bounded by the curve

x=e^{y/2}

the $y$ -axis and the horizontal lines $y=0$ and $y=\ln4$ . The region is rotated through $2\pi$ radians about the $y$ -axis.

A

Write down an integral representing the volume of the solid formed.

[2]

B

Calculate the exact volume of the solid.

[3]

Question 16

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The curve $x=y^2-4$ and the $y$ -axis enclose a finite region.

Sideways parabola x = y^2 - 4 and the y-axis.

A

Find the $y$ -coordinates of the points where the curve intersects the $y$ -axis.

[1]

B

Write down a definite integral for the signed quantity obtained by integrating $x$ with respect to $y$ between these points.

[1]

C

Hence find the area of the enclosed region.

[3]

Question 17

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The height of a decorative canopy above a horizontal walkway is modelled by

h(x)=-0.02x^3+0.18x^2+0.5

where $0\leq x\leq 8$ , and $x$ and $h$ are measured in metres.

A

I.

Find an antiderivative of $h(x)$ .

[2]

II.

Calculate the area of the vertical cross-section beneath the canopy.

[2]

B

I.

Use the trapezoidal rule with four equal intervals to estimate the cross-sectional area.

[3]

II.

Find the percentage error in the trapezoidal estimate and state whether it is an overestimate or an underestimate. If you did not obtain an area in part (a)(ii), use $14.24\ \text{m}^2$ .

[3]

Question 18

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

During a fundraising event, the total amount collected, in hundreds of dollars, is $Q(t)$ , where $t$ is the number of hours after the event begins. The rate of collection is modelled by

\frac{dQ}{dt}=0.3t^2-1.8t+4.5

for $0\leq t\leq6$ . At the start of the event, $Q=25$ .

A

I.

Find the general form of $Q(t)$ .

[2]

II.

Use the initial condition to determine a formula for $Q(t)$ .

[2]

B

I.

Calculate the amount collected by the end of the six hours, giving your answer in dollars.

[2]

II.

Find the average rate of collection during the six-hour event, in dollars per hour. If you did not obtain $Q(6)$ , use $41.2$ hundreds of dollars.

[2]

C

Determine when the amount collected first reaches $3500$ dollars.

[2]

Question 19

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The vertical cross-section of a theatre backdrop is bounded by the $x$ -axis, the lines $x=0$ and $x=20$ , and the curve

$y=-0.04x^2+0.8x+1$

where both coordinates are measured in metres.

A

I.

Write down a definite integral representing the area of the backdrop.

[1]

II.

Find the area of the backdrop.

[3]

B

I.

Use the trapezoidal rule with four equal intervals to estimate the area.

[2]

II.

Calculate the percentage error in this estimate. If you did not obtain the exact area, use $73.333\ldots\ \text{m}^2$ .

[2]

C

Painting the backdrop costs 18.50 dollars per square metre. Find the total cost, to the nearest dollar, using the exact area from part (a).

[2]

Question 20

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The electrical power produced by a small wind turbine is recorded every $0.5$ hours during a three-hour test. The results are shown in the table.

Time [h]	Power [kW]
0.0	1.2
0.5	1.8
1.0	2.5
1.5	3.1
2.0	2.7
2.5	2.0
3.0	1.4

A

I.

The recorded powers, in order, are $1.2$ , $1.8$ , $2.5$ , $3.1$ , $2.7$ , $2.0$ and $1.4$ kilowatts. Use the trapezoidal rule to estimate the energy produced.

[3]

II.

State why the units of the estimate are kilowatt-hours rather than kilowatts.

[1]

B

I.

A battery has a stated capacity of $8\ \text{kWh}$ , of which $85\%$ is usable. Determine whether the usable capacity is sufficient to store the estimated energy.

[2]

II.

Explain one limitation of using the trapezoidal estimate to select the battery.

[1]

C

An alternative model for the power is $p(t)=-0.4t^2+1.2t+1.3$ . Calculate the energy predicted by this model for $0\leq t\leq3$ and compare it with the trapezoidal estimate.

[2]

Question 21

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A survey team measures the width of a coastal wetland at $40$ metre intervals along a baseline of length $240$ metres. The measured widths are shown in the table.

Distance along baseline [m]	Wetland width [m]
0	0
40	22
80	35
120	41
160	33
200	18
240	0

A

I.

State the number of intervals and their common width.

[1]

II.

The recorded widths, in order, are $0$ , $22$ , $35$ , $41$ , $33$ , $18$ and $0$ metres. Use the trapezoidal rule to estimate the area of the wetland.

[3]

B

I.

A satellite model estimates the wetland area to be $5720\ \text{m}^2$ . Calculate the percentage by which the trapezoidal estimate exceeds the satellite estimate.

[2]

II.

Suggest one reason why the two estimates may differ.

[1]

C

The two endpoint widths are known exactly, but each interior width may have an error of up to $2\ \text{m}$ . Find the resulting lower and upper bounds for the trapezoidal estimate.

[3]

Question 22

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A manufacturer cuts a panel whose height is modelled by

p(x)=0.001x^3-0.06x^2+0.9x+2

for $0\leq x\leq20$ , where both $x$ and $p(x)$ are measured in metres. The panel lies between the curve and the $x$ -axis. For part (b), a vertical cut at $x=a$ divides the panel into two regions of equal area.

A

I.

Find an antiderivative of $p(x)$ .

[2]

II.

Calculate the area of the panel.

[2]

B

A vertical cut at $x=a$ is to divide the panel into two pieces of equal area.

I.

Write down an equation that can be solved to find $a$ .

[1]

II.

Hence determine the value of $a$ .

[2]

C

I.

The manufactured panel is vertically scaled by a factor of $0.95$ , while its width is unchanged. Find the area of the scaled panel.

[2]

II.

The material costs $12.40$ dollars per square metre. Calculate the cost of the scaled panel.

[2]

Question 23

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The number of people in an exhibition hall is $N(t)$ , where $t$ is measured in minutes after the doors open. The net rate of change is modelled by

\frac{dN}{dt}=-0.02t^3+0.3t^2+2

for $0\leq t\leq10$ . Initially, there are $120$ people in the hall.

A

I.

Find a formula for $N(t)$ .

[3]

II.

Find the number of people in the hall after $10$ minutes.

[1]

B

I.

Determine when the number of people first reaches $160$ .

[2]

II.

Find the average net rate of change in the number of people during the first $10$ minutes. If you did not obtain $N(10)$ , use $190$ .

[1]

C

Justify why the model predicts that the number of people is increasing throughout the stated interval.

[2]

Question 24

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A curve has gradient

$\frac{dy}{dx}=6x-2$

and passes through the point $(0,2)$ . The region between the curve, the $x$ -axis and the lines $x=0$ and $x=2$ is denoted by $R$ .

Upward-opening curve on 0≤x≤2 with sample ordinates.

A

I.

Find the equation of the curve.

[3]

II.

Show that the curve lies above the $x$ -axis for all real values of $x$ .

[1]

B

Find the area of $R$ .

[3]

C

I.

Use the trapezoidal rule with four equal intervals to estimate the area of $R$ .

[2]

II.

Explain why the trapezoidal estimate is greater than the exact area.

[1]

Question 25

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A treatment system removes a contaminant from a vessel at the rate

$r(t)=40t e^{-t^2/4}$

grams per hour, where $t\geq0$ . Initially, the vessel contains $120$ grams of the contaminant.

A

I.

Using the substitution $u=-\frac{t^2}{4}$ , find an antiderivative of $r(t)$ .

[3]

II.

Show that the mass removed during the first $T$ hours is

$80\left(1-e^{-T^2/4}\right)$

[1]

B

I.

Calculate the mass remaining after two hours.

[2]

II.

State the limiting mass remaining according to the model as $t\to\infty$ .

[1]

C

Determine when the mass remaining first reaches $60$ grams.

[3]

Question 26

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The profile of a rotationally symmetric wooden ornament is modelled by

$y=2+\sqrt{x},\qquad 0\leq x\leq9$

where $x$ and $y$ are measured in centimetres. The region under the curve is rotated through $2\pi$ radians about the $x$ -axis.

A

I.

Write down an integral representing the volume of the ornament.

[2]

II.

Expand the integrand and find an antiderivative.

[2]

B

Hence find the volume of the ornament, giving both an exact answer and an answer to three significant figures.

[2]

C

A cylindrical hole of radius $1\ \text{cm}$ is drilled along the $x$ -axis through the entire $9\ \text{cm}$ length of the ornament, centred on the axis of rotation.

I.

Find the volume remaining after the hole is drilled.

[2]

II.

Calculate the percentage of the original volume removed.

[2]

Question 27

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A function $F$ is defined for $x>0$ and satisfies

$F'(x)=x^{-1/2}+\frac{3}{x}$

It is given that $F(1)=2$ .

A

I.

Find the general form of $F(x)$ .

[2]

II.

Use the boundary condition to determine $F(x)$ .

[2]

B

I.

Find the exact value of $\displaystyle\int_1^4F'(x)\,dx$ .

[2]

II.

Hence find the average value of $F'(x)$ on $1\leq x\leq4$ .

[1]

C

Determine the value of $x$ for which $F(x)=10$ .

[2]

Question 28

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A filtration system removes a contaminant at a rate
$r(t)=12e^{-0.3t}+3$
where $r$ is measured in milligrams per minute and $t$ is the time, in minutes, after the system starts. The term $3$ represents a constant background removal rate.

Removal rate model with horizontal background rate 3.

A

I.

Find an antiderivative of $r(t)$ .

[2]

II.

Hence show that the mass removed during the first $T$ minutes is
$M(T)=40(1-e^{-0.3T})+3T$

[2]

B

Calculate the mass removed during the first $8$ minutes.

[2]

C

I.

Find the total mass that will eventually be removed by the exponentially decreasing component of the rate.

[2]

II.

Determine when $90\%$ of the mass in part (c)(i) has been removed. If you did not obtain $40$ in part (c)(i), use $40$ .

[2]

III.

Explain why the total mass removed by the complete rate model does not approach a finite limit.

[2]

Question 29

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The rate of change of the volume of water in a tidal basin is modelled by
$q(t)=6\sin\left(\frac{\pi t}{6}\right),\qquad 0\leq t\leq12$
where $q$ is measured in thousands of cubic metres per hour. Positive values represent inflow and negative values represent outflow.

Graph of q(t) over one full 12-hour sinusoidal cycle, showing inflow for positive values and outflow for negative values.

A

I.

Find an antiderivative of $q(t)$ .

[2]

II.

Calculate the signed change in volume over the $12$ hours.

[2]

B

Find the total volume of water that passes through the basin entrance, counting inflow and outflow as positive.

[3]

C

I.

Show that the increase in the basin volume from time $0$ to time $t$ is
$A(t)=\frac{36}{\pi}\left(1-\cos\left(\frac{\pi t}{6}\right)\right)$

[2]

II.

Determine the two times when the basin contains half of its maximum additional volume. If you did not obtain $A(t)$ , use the expression given in part (c)(i).

[3]

Question 30

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The inside profile of a rotationally symmetric vessel is modelled by
$x=0.8+0.12y^2,\qquad 0\leq y\leq5$
where $x$ and $y$ are measured in metres. The region between the profile and the $y$ -axis is rotated through $2\pi$ radians about the $y$ -axis.

A

I.

Write down an integral for the volume of the vessel.

[2]

II.

Find the exact volume and its value to three significant figures.

[3]

B

A cylindrical vessel of height $5$ metres has the same volume. Determine its radius.

[3]

C

A second vessel has profile $x=a+0.1y^2$ for $0\leq y\leq4$ and volume $10\pi\ \text{m}^3$ . Determine the positive value of $a$ .

[4]

Question 31

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The accumulated quantity of a dissolved mineral is modelled from the rate
$r(t)=\frac{18t}{t^2+4},\qquad t\geq0$
where $r$ is measured in grams per hour.

Graph of the accumulation rate r(t) against time t.

A

I.

Using the substitution $u=t^2+4$ , find an antiderivative of $r(t)$ .

[2]

II.

Show that the quantity accumulated by time $T$ is
$Q(T)=9\ln\left(\frac{T^2+4}{4}\right)$

[2]

B

Determine when the accumulated quantity first reaches $20$ grams.

[3]

C

I.

Find the average rate of accumulation from $t=0$ until the time found in part (b). If you did not obtain a time, use $5.74$ hours.

[2]

II.

Determine whether $Q(T)$ approaches a finite limit as $T\to\infty$ and justify your answer.

[3]

Question 32

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The electrical power generated by a solar installation is modelled by
$P(t)=5+3\cos\left(\frac{\pi(t-2)}6\right),\qquad0\leq t\leq12$
where $P$ is measured in kilowatts and $t$ is measured in hours.

Power vs time graph for a solar installation over 12 hours with a 5 kW baseline.

A

I.

Find the total electrical energy generated during the $12$ hours.

[2]

II.

State the average power over this interval.

[2]

B

Find the intervals during which the generated power exceeds $5\ \text{kW}$ .

[3]

C

I.

Calculate the energy generated above the baseline power of $5\ \text{kW}$ .

[3]

II.

Hence find the average power during the times when the power exceeds $5\ \text{kW}$ .

[2]

Question 33

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The curve

$y=2\sin x-1$

is considered on the interval $0\leq x\leq2\pi$ .

A

I.

Find the two values of $x$ at which the curve intersects the $x$ -axis.

[2]

II.

State the interval on which the curve is above the $x$ -axis.

[1]

B

I.

Find the signed integral $\displaystyle\int_0^{2\pi}(2\sin x-1)\,dx$ .

[2]

II.

Explain why the answer to part (b)(i) is not the total geometric area.

[2]

C

Find the total geometric area between the curve and the $x$ -axis for $0\leq x\leq2\pi$ .

[4]

Question 34

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A designer models the outer profile of a lampshade by

$x=1+2e^{-y/4},\qquad 0\leq y\leq8$

where $x$ and $y$ are measured in centimetres. The region between the curve and the $y$ -axis is rotated through $2\pi$ radians about the $y$ -axis.

A

I.

Write down an integral representing the volume of the lampshade model.

[2]

II.

Show that the integrand can be written as $1+4e^{-y/4}+4e^{-y/2}$ .

[2]

B

Hence calculate the volume of the model.

[3]

C

I.

A cylindrical model has radius $3\ \text{cm}$ and height $8\ \text{cm}$ . Find its volume.

[1]

II.

Calculate the percentage by which the exponential-profile volume is less than the cylindrical volume.

[2]

Question 35

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The curve

$y=x^3-4x$

is considered on the interval $-3\leq x\leq3$ .

A

I.

Find the $x$ -intercepts of the curve.

[2]

II.

State the symmetry of the graph.

[1]

B

I.

Find $\displaystyle\int_{-3}^{3}(x^3-4x)\,dx$ .

[1]

II.

Explain why this value does not imply that the geometric area is zero.

[2]

C

Find the total geometric area between the curve and the $x$ -axis for $-3\leq x\leq3$ .

[4]

Question 36

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $0\leq x\leq\pi$ , define

$f(x)=\frac{4\sin x}{3-\cos x}$

The accumulated value from $0$ to $b$ is denoted by

$A(b)=\int_0^b f(x)\,dx$

A

I.

Using the substitution $u=3-\cos x$ , find an antiderivative of $f(x)$ .

[3]

II.

Find the exact value of $A(\pi)$ .

[1]

B

Let $b$ be a value in $0\leq b\leq\pi$ such that $A(b)=\frac12A(\pi)$ .

I.

Show that $\cos b=3-2\sqrt2$ .

[3]

II.

Hence find $b$ .

[1]

C

Explain why the value of $b$ in part (b) is unique on $0\leq b\leq\pi$ .

[2]

Question 37

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A decorative solid is formed by rotating the region under
$y=\sqrt{x}e^{-x/2},\qquad 0\leq x\leq a$
about the $x$ -axis, where $a>0$ .

A

I.

Show that the volume is
$V(a)=\pi\int_0^a xe^{-x}\,dx$

[2]

II.

Use technology or an appropriate antiderivative to show that
$V(a)=\pi\left(1-(a+1)e^{-a}\right)$

[3]

B

Calculate the volume when $a=4$ .

[2]

C

I.

State the limiting volume as $a\to\infty$ .

[1]

II.

Determine the value of $a$ for which the volume is $95\%$ of its limiting value. If you did not obtain the limiting volume, use $\pi$ .

[3]

III.

Explain how a solid extending indefinitely in the positive $x$ -direction can have a finite volume.

[1]

Question 38

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The value of
$I=\int_0^2e^{-x^2}\,dx$
is approximated using the trapezoidal rule. Values of $e^{-x^2}$ are generated using a GDC.

$x$	$e^{-x^2}$
0.00	1.000000
0.25	0.939413
0.50	0.778801
0.75	0.569783
1.00	0.367879
1.25	0.209611
1.50	0.105399
1.75	0.046771
2.00	0.018316

A

I.

Use four equal intervals to find the trapezoidal estimate $T_4$ .

[3]

II.

Use eight equal intervals to find the trapezoidal estimate $T_8$ .

[2]

B

A high-accuracy numerical integration gives $I=0.882081$ . Calculate the percentage error in $T_8$ .

[2]

C

I.

The leading trapezoidal error is assumed to be proportional to $h^2$ . Show that reducing the interval width by a factor of $2$ should reduce this error by a factor of $4$ .

[2]

II.

Hence derive and use an improved estimate
$I\approx T_8+\frac{T_8-T_4}{3}$

[3]

Question 39

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The curve
$x=y^3-4y$
and the $y$ -axis enclose two finite regions.

Curve x = y^3 - 4y with its y-axis intersections.

A

I.

Find the $y$ -coordinates of the intersections with the $y$ -axis.

[2]

II.

Find the signed integral $\int_{-2}^{2}(y^3-4y)\,dy$ and explain its value.

[3]

B

Find the total geometric area enclosed by the curve and the $y$ -axis.

[3]

C

I.

The two regions are rotated about the $y$ -axis. Find the exact total volume formed.

[3]

II.

For the family $x=y^3-a^2y$ , where $a>0$ , the corresponding total enclosed area is $A$ . Deduce the volume of revolution in terms of $A$ .

[2]

Question 40

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A storage tank is modelled by rotating the region under
$y=k\left(1-\frac{x^2}{a^2}\right),\qquad -a\leq x\leq a$
about the $x$ -axis, where $a>0$ and $k>0$ .

A

I.

Find the area $A$ of the generating region.

[2]

II.

Show that the tank volume is
$V=\frac{16\pi ak^2}{15}$

[3]

B

For a tank with $a=3$ and $k=3$ , calculate its volume.

[2]

C

I.

For tanks with a fixed generating area $A$ , express $V$ in terms of $A$ and $a$ .

[3]

II.

Interpret how the volume changes as $a$ increases while $A$ remains fixed.

[2]

Question 41

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The rate at which a medicine is absorbed is modelled by

r(t)=At e^{-kt^2},\qquad t\geq0

where $t$ is measured in hours, $r$ is measured in milligrams per hour, $k=0.5\ \text{h}^{-2}$ , and $A>0$ has units $\text{mg h}^{-2}$ . The total amount eventually absorbed is $80$ milligrams.

Absorption rate of the medicine over time.

A

I.

Show that the total amount eventually absorbed is $\frac{A}{2k}$ milligrams.

[3]

II.

Hence state the value of $A$ , including its units.

[2]

B

Determine the time by which $75\%$ of the medicine has been absorbed.

[3]

C

I.

Use the trapezoidal rule with interval width $0.5$ hours to estimate the amount absorbed during the first $2$ hours.

[2]

II.

The exact amount absorbed during the first $2$ hours is $80(1-e^{-2})$ . Calculate the percentage error in the trapezoidal estimate and state its direction.

[2]

Question 42

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The vertical width of a proposed tunnel was measured at horizontal positions $x=0$ , $2.5$ , $5$ , $7.5$ and $10$ metres. The corresponding widths were $0$ , $4.6$ , $6.1$ , $4.7$ and $0$ metres. A mathematical model for the width is
$w(x)=kx(10-x),\qquad0\leq x\leq10$

x [m]	width [m]
0	0
2.5	4.6
5	6.1
7.5	4.7
10	0

A

I.

Use the trapezoidal rule to estimate the cross-sectional area from the measurements.

[2]

II.

Use the central measurement to determine $k$ .

[2]

B

Find the cross-sectional area predicted by the model and compare it with the trapezoidal estimate.

[3]

C

I.

The model region is rotated about the $x$ -axis. Find the volume of the resulting solid.

[3]

II.

For the general model $w(x)=kx(L-x)$ on $0\leq x\leq L$ , show that its volume of revolution $V$ and cross-sectional area $A$ satisfy
$V=\frac{6\pi A^2}{5L}$

[3]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $c>1$ , consider the positive function
$f_c(x)=\frac{\sin x}{c+\cos x},\qquad0\leq x\leq\pi$

Positive family of f_c(x) on [0,π] for several c>1.

A

I.

Using the substitution $u=c+\cos x$ , find an antiderivative of $f_c(x)$ .

[2]

II.

Show that the area under the curve is
$A(c)=\ln\left(\frac{c+1}{c-1}\right)$

[3]

B

Determine $c$ if the area under the curve is $\ln2$ .

[2]

C

I.

For $c=2$ , find the value $a$ such that the area from $x=0$ to $x=a$ is half the total area.

[3]

II.

State what happens to $A(c)$ as $c\to\infty$ .

[1]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The difference between two decaying concentrations is modelled by
$d(x)=e^{-x}-e^{-2x},\qquad x\geq0$

A

I.

Show that $d(x)\geq0$ for $x\geq0$ .

[2]

II.

Find the total area under $d(x)$ for $x\geq0$ .

[2]

B

Show that the area accumulated from $0$ to $T$ is
$A(T)=\frac12(1-e^{-T})^2$

[3]

C

I.

Determine when $90\%$ of the total area has accumulated.

[3]

II.

The entire region is rotated about the $x$ -axis. Find the exact volume formed.

[2]

Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $0<a<4$ , the curve
$y=x^2-a$
is considered on the interval $-2\leq x\leq2$ .

Curve	Parameter range	$x$ interval
$y=x^2-a$	$0<a<4$	$-2\leq x\leq2$

A

I.

Find the signed integral $S(a)=\int_{-2}^{2}(x^2-a)\,dx$ .

[2]

II.

Determine the value of $a$ for which the signed integral is zero.

[3]

B

Show that the total geometric area is
$G(a)=\frac{16}{3}-4a+\frac{8}{3}a^{3/2}$

[4]

C

I.

Find the geometric area when the signed integral is zero.

[2]

II.

For this same value of $a$ , the region between the curve and the $x$ -axis is rotated about the $x$ -axis. Find the exact volume formed.

[2]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $h>0$ , a region is bounded by the $y$ -axis and
$x=\sqrt{y(h-y)},\qquad0\leq y\leq h$

A

I.

Show that the curve is part of a circle and state its radius.

[2]

II.

Hence find the area $A$ of the region.

[2]

B

The region is rotated about the $y$ -axis. Show that its volume is
$V=\frac{\pi h^3}{6}$

[3]

C

I.

Find $A$ and $V$ when $h=6$ .

[2]

II.

Eliminate $h$ to express $V$ in terms of $A$ .

[2]

Question 47

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The profile of a rotationally symmetric component is modelled in two sections by

$y=1+\frac{x}{2},\qquad 0\leq x\leq2$

and

$y=2\cos\left(\frac{\pi(x-2)}{6}\right),\qquad 2\leq x\leq5$

The coordinates are measured in centimetres. The region under the profile is rotated through $2\pi$ radians about the $x$ -axis.

A

I.

Verify that the two sections meet at $x=2$ .

[2]

II.

State the radius of the component at $x=5$ .

[1]

B

I.

Write down the volume as the sum of two definite integrals.

[3]

II.

Evaluate the volume of the first section exactly.

[1]

III.

Evaluate the volume of the second section exactly. You may use $\cos^2u=\frac12(1+\cos2u)$ .

[2]

C

Hence find the total volume of the component, giving an exact answer and an answer to three significant figures.

[2]

D

The second section is replaced by a cone of base radius $2\ \text{cm}$ and height $3\ \text{cm}$ . Calculate the percentage by which the original second-section volume exceeds the cone's volume.

[1]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The integral
$I=\int_0^1e^x\,dx$
is approximated using the trapezoidal rule with $N$ equal intervals. The resulting estimate is denoted by $T_N$ .

Graph of y=e^x on [0,1] with trapezoidal chords for N=4 and N=8.

A

I.

Find the exact value of $I$ .

[1]

II.

Find $T_4$ and $T_8$ .

[4]

B

Explain why both estimates in part (a)(ii) are greater than the exact value.

[2]

C

I.

Show that
$T_N=\frac1N\left[\frac{1+e}{2}+\frac{e-e^{1/N}}{e^{1/N}-1}\right]$

[3]

II.

Using the expression in part (c)(i), determine the least value of $N$ for which the percentage error in $T_N$ is less than $0.01\%$ .

[3]