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AHL 5.17—Areas under curve onto y-axis, volume of revolution (about x and y axes) - IB Questionbank | RevisionDojo

Practice AHL 5.17—Areas under curve onto y-axis, volume of revolution (about x and y axes) with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Let $f(x)=4-x^2$ , where $x \geq 0$ . The shaded region is enclosed by the graph of $f$ , the $x$ -axis, and the $y$ -axis.

Find the $x$ -coordinate of the point where the graph of $f$ intersects the $x$ -axis.

[2]

Find the area of the shaded region.

[2]

Find the volume of the solid formed when the shaded region is revolved $360^{\circ}$ about the $x$ -axis.

[4]

Question 2

HLPaper 2

Let $f(x)=\frac{1}{x^{2}+2}$ , for $0 \leq x \leq 2$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x -axis, and the lines $x=0$ and $x=2$ .

Sketch the graph of $y=f(x)$ for $0 \leq x \leq 2$ , indicating the intercepts and the maximum point.

[3]

Find the area of region $R$ , giving your answer in the form $\frac{\pi a}{\sqrt{b}}$ , where $a, b \in \mathbb{Z}$ .

[4]

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

[5]

Question 3

HLPaper 2

Consider the function $g(x) = \frac{\sqrt x }{\sin x}$ , $0 < x < \pi$ .

Consider the region bounded by the curve $y = g(x)$ , the $x$ -axis and the lines $x = \frac{\pi }{6}, x = \frac{\pi }{3}$ .

This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.

[3]

Sketch the graph of $y = g(x)$ showing clearly the minimum point and any asymptotic behaviour.

[3]

Determine the values of $x$ for which $g(x)$ is a decreasing function.

[2]

Show that the $x$ -coordinate of the minimum point on the curve $y = g(x)$ satisfies the equation $\tan(x) = 2x$ .

[5]

Find the coordinates of the point on the graph of $g$ where the normal to the graph is parallel to the line $y = -x$ .

[4]

Question 4

HLPaper 2

Let $f(x)=e^{-x} \sin x$ , for $0 \leq x \leq \pi$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=0$ and $x=\pi$ .

Sketch the graph of $y=f(x)$ , indicating the coordinates of the maximum point and intercepts.

[3]

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

[5]

Question 5

HLPaper 2

All lengths in this question are in metres. Consider the function $f(x)=\frac{4}{x^{2}+4}$ , for $-1 \leq x \leq 1$ . The region $R$ is enclosed by the graph of $f$ , the x -axis, and the lines $x=-1$ and $x=1$ . A container is modelled by rotating this region $360^{\circ}$ about the x-axis.

The volume of water in the container is given by the function $g(t)$ , for $0 \leq t \leq 3$ , where $t$ is measured in hours and $g(t)$ is in $\mathrm{m}^{3}$ . The rate of change of the volume is $g^{\prime}(t)=0.8-1.5 \sin (0.5 t)$ . The volume of water is increasing only when $p<t<q$ .

Find the volume of the container.

[3]

Find the values of $p$ and $q$ .

[3]

During the interval $p<t<q$ , the volume of water increases by $k \mathrm{~m}^{3}$ . Find the value of $k$ .

[3]

When $t=0$ , the volume of water is $0.5 \mathrm{~m}^{3}$ . Find the minimum volume of empty space in the container during the 3 -hour period, given the container is never full.

[5]

Question 6

HLPaper 1

Consider $f(x)=\frac{\ln x}{x}$ , for $1 \leq x \leq e^{2}$ . Let $R$ be the region bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=1$ and $x=e^{2}$ .

Find the area of region $R$ .

[4]

Find the volume of the solid formed when region $R$ is rotated $2 \pi$ radians about the y-axis, giving your answer in terms of $\pi$ .

[4]

Question 7

HLPaper 2

Let $f(x)=x \ln x$ , for $1 \leq x \leq e$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=1$ and $x=e$ .

Sketch the graph of $y=f(x)$ , indicating the intercepts and the point where $x=e$ .

[2]

Find the area of region $R$ .

[4]

Find the volume of the solid formed when region $R$ is rotated $2 \pi$ radians about the x-axis, giving your answer in terms of $\pi$ .

[5]

Question 8

HLPaper 2

Let $f(x)=\frac{2 x}{x^{2}+1}$ , for $x \in \mathbb{R}$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=0$ and $x=2$ .

Sketch the graph of $y=f(x)$ for $0 \leq x \leq 2$ , clearly indicating the coordinates of any intercepts and the maximum point.

[3]

Find the area of region $R$ .

[3]

The region $R$ is rotated $2 \pi$ radians about the x -axis to form a solid of revolution. Find the volume of the solid formed, giving your answer in the form $\pi(a+b \ln c)$ , where $a, b, c \in \mathbb{Z}$ .

[6]

Question 9

HLPaper 1

Consider the function $f(x) = x^2 + 2x + 1$ on the interval $[0, 2]$ .

Calculate the volume of the solid formed by revolving the curve $f(x)$ around the $x$ -axis from $x = 0$ to $x = 2$ .

[6]

Question 10

HLPaper 1

Let $g(x)=\sqrt{9-x^2}$ , for $-3 \leq x \leq 3$ . The graph represents the upper half of a circle with radius 3, centered at the origin.

Find the area enclosed by the graph of $g(x)$ , the $x$ -axis, and the limits $x=-3$ and $x=3$ .

[4]

Find the volume of the solid formed when the region is revolved $360^{\circ}$ about the $x$ -axis using integration.

[6]

Paper

Level

Question 1

HLPaper 1

Let $f(x)=4-x^2$ , where $x \geq 0$ . The shaded region is enclosed by the graph of $f$ , the $x$ -axis, and the $y$ -axis.

Find the $x$ -coordinate of the point where the graph of $f$ intersects the $x$ -axis.

[2]

Find the area of the shaded region.

[2]

Find the volume of the solid formed when the shaded region is revolved $360^{\circ}$ about the $x$ -axis.

[4]

Question 2

HLPaper 2

Let $f(x)=\frac{1}{x^{2}+2}$ , for $0 \leq x \leq 2$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x -axis, and the lines $x=0$ and $x=2$ .

Sketch the graph of $y=f(x)$ for $0 \leq x \leq 2$ , indicating the intercepts and the maximum point.

[3]

Find the area of region $R$ , giving your answer in the form $\frac{\pi a}{\sqrt{b}}$ , where $a, b \in \mathbb{Z}$ .

[4]

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

[5]

Question 3

HLPaper 2

Consider the function $g(x) = \frac{\sqrt x }{\sin x}$ , $0 < x < \pi$ .

Consider the region bounded by the curve $y = g(x)$ , the $x$ -axis and the lines $x = \frac{\pi }{6}, x = \frac{\pi }{3}$ .

This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.

[3]

Sketch the graph of $y = g(x)$ showing clearly the minimum point and any asymptotic behaviour.

[3]

Determine the values of $x$ for which $g(x)$ is a decreasing function.

[2]

Show that the $x$ -coordinate of the minimum point on the curve $y = g(x)$ satisfies the equation $\tan(x) = 2x$ .

[5]

Find the coordinates of the point on the graph of $g$ where the normal to the graph is parallel to the line $y = -x$ .

[4]

Question 4

HLPaper 2

Let $f(x)=e^{-x} \sin x$ , for $0 \leq x \leq \pi$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=0$ and $x=\pi$ .

Sketch the graph of $y=f(x)$ , indicating the coordinates of the maximum point and intercepts.

[3]

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

[5]

Question 5

HLPaper 2

Find the volume of the container.

[3]

Find the values of $p$ and $q$ .

[3]

During the interval $p<t<q$ , the volume of water increases by $k \mathrm{~m}^{3}$ . Find the value of $k$ .

[3]

When $t=0$ , the volume of water is $0.5 \mathrm{~m}^{3}$ . Find the minimum volume of empty space in the container during the 3 -hour period, given the container is never full.

[5]

Question 6

HLPaper 1

Consider $f(x)=\frac{\ln x}{x}$ , for $1 \leq x \leq e^{2}$ . Let $R$ be the region bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=1$ and $x=e^{2}$ .

Find the area of region $R$ .

[4]

Find the volume of the solid formed when region $R$ is rotated $2 \pi$ radians about the y-axis, giving your answer in terms of $\pi$ .

[4]

Question 7

HLPaper 2

Let $f(x)=x \ln x$ , for $1 \leq x \leq e$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=1$ and $x=e$ .

Sketch the graph of $y=f(x)$ , indicating the intercepts and the point where $x=e$ .

[2]

Find the area of region $R$ .

[4]

Find the volume of the solid formed when region $R$ is rotated $2 \pi$ radians about the x-axis, giving your answer in terms of $\pi$ .

[5]

Question 8

HLPaper 2

Let $f(x)=\frac{2 x}{x^{2}+1}$ , for $x \in \mathbb{R}$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=0$ and $x=2$ .

Sketch the graph of $y=f(x)$ for $0 \leq x \leq 2$ , clearly indicating the coordinates of any intercepts and the maximum point.

[3]

Find the area of region $R$ .

[3]

[6]

Question 9

HLPaper 1

Consider the function $f(x) = x^2 + 2x + 1$ on the interval $[0, 2]$ .

Calculate the volume of the solid formed by revolving the curve $f(x)$ around the $x$ -axis from $x = 0$ to $x = 2$ .

[6]

Question 10

HLPaper 1

Let $g(x)=\sqrt{9-x^2}$ , for $-3 \leq x \leq 3$ . The graph represents the upper half of a circle with radius 3, centered at the origin.

Find the area enclosed by the graph of $g(x)$ , the $x$ -axis, and the limits $x=-3$ and $x=3$ .

[4]

Find the volume of the solid formed when the region is revolved $360^{\circ}$ about the $x$ -axis using integration.

[6]

Paper

Level

Question 1

HLPaper 1

Let $f(x)=4-x^2$ , where $x \geq 0$ . The shaded region is enclosed by the graph of $f$ , the $x$ -axis, and the $y$ -axis.

Find the $x$ -coordinate of the point where the graph of $f$ intersects the $x$ -axis.

[2]

Find the area of the shaded region.

[2]

Find the volume of the solid formed when the shaded region is revolved $360^{\circ}$ about the $x$ -axis.

[4]

Question 2

HLPaper 2

Let $f(x)=\frac{1}{x^{2}+2}$ , for $0 \leq x \leq 2$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x -axis, and the lines $x=0$ and $x=2$ .

Sketch the graph of $y=f(x)$ for $0 \leq x \leq 2$ , indicating the intercepts and the maximum point.

[3]

Find the area of region $R$ , giving your answer in the form $\frac{\pi a}{\sqrt{b}}$ , where $a, b \in \mathbb{Z}$ .

[4]

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

[5]

Question 3

HLPaper 2

Consider the function $g(x) = \frac{\sqrt x }{\sin x}$ , $0 < x < \pi$ .

Consider the region bounded by the curve $y = g(x)$ , the $x$ -axis and the lines $x = \frac{\pi }{6}, x = \frac{\pi }{3}$ .

This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.

[3]

Sketch the graph of $y = g(x)$ showing clearly the minimum point and any asymptotic behaviour.

[3]

Determine the values of $x$ for which $g(x)$ is a decreasing function.

[2]

Show that the $x$ -coordinate of the minimum point on the curve $y = g(x)$ satisfies the equation $\tan(x) = 2x$ .

[5]

Find the coordinates of the point on the graph of $g$ where the normal to the graph is parallel to the line $y = -x$ .

[4]

Question 4

HLPaper 2

Let $f(x)=e^{-x} \sin x$ , for $0 \leq x \leq \pi$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=0$ and $x=\pi$ .

Sketch the graph of $y=f(x)$ , indicating the coordinates of the maximum point and intercepts.

[3]

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

[5]

Question 5

HLPaper 2

Find the volume of the container.

[3]

Find the values of $p$ and $q$ .

[3]

During the interval $p<t<q$ , the volume of water increases by $k \mathrm{~m}^{3}$ . Find the value of $k$ .

[3]

When $t=0$ , the volume of water is $0.5 \mathrm{~m}^{3}$ . Find the minimum volume of empty space in the container during the 3 -hour period, given the container is never full.

[5]

Question 6

HLPaper 1

Consider $f(x)=\frac{\ln x}{x}$ , for $1 \leq x \leq e^{2}$ . Let $R$ be the region bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=1$ and $x=e^{2}$ .

Find the area of region $R$ .

[4]

Find the volume of the solid formed when region $R$ is rotated $2 \pi$ radians about the y-axis, giving your answer in terms of $\pi$ .

[4]

Question 7

HLPaper 2

Let $f(x)=x \ln x$ , for $1 \leq x \leq e$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=1$ and $x=e$ .

Sketch the graph of $y=f(x)$ , indicating the intercepts and the point where $x=e$ .

[2]

Find the area of region $R$ .

[4]

Find the volume of the solid formed when region $R$ is rotated $2 \pi$ radians about the x-axis, giving your answer in terms of $\pi$ .

[5]

Question 8

HLPaper 2

Let $f(x)=\frac{2 x}{x^{2}+1}$ , for $x \in \mathbb{R}$ . Consider the region $R$ bounded by the graph of $y=f(x)$ , the x-axis, and the lines $x=0$ and $x=2$ .

Sketch the graph of $y=f(x)$ for $0 \leq x \leq 2$ , clearly indicating the coordinates of any intercepts and the maximum point.

[3]

Find the area of region $R$ .

[3]

[6]

Question 9

HLPaper 1

Consider the function $f(x) = x^2 + 2x + 1$ on the interval $[0, 2]$ .

Calculate the volume of the solid formed by revolving the curve $f(x)$ around the $x$ -axis from $x = 0$ to $x = 2$ .

[6]

Question 10

HLPaper 1

Let $g(x)=\sqrt{9-x^2}$ , for $-3 \leq x \leq 3$ . The graph represents the upper half of a circle with radius 3, centered at the origin.

Find the area enclosed by the graph of $g(x)$ , the $x$ -axis, and the limits $x=-3$ and $x=3$ .

[4]

Find the volume of the solid formed when the region is revolved $360^{\circ}$ about the $x$ -axis using integration.

[6]

AHL 5.17—Areas under curve onto y-axis, volume of revolution (about x and y axes) Questionbank