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AHL 5.12—Areas under a curve onto x or y axis. Volumes of revolution about x and y - IB Questionbank | RevisionDojo

Practice AHL 5.12—Areas under a curve onto x or y axis. Volumes of revolution about x and y with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 2

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a = 3{t^2} - 14t + 8$ , for $0 \leqslant t \leqslant 5$ .

When $t = 0$ , the velocity of P is $3{\text{ m}}\,{{\text{s}}^{ - 1}}$ .

Write down the values of $t$ when $a = 0$ .

[2]

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

[2]

Find an expression for the velocity of P at time $t$ .

[6]

Find the total distance travelled by P when its velocity is increasing.

[4]

Question 2

HLPaper 2

The function $x = \sqrt{4 - y}$ is defined for $0 \leq y \leq 4$ . The shaded region is enclosed by the curve, the $y$ -axis, and the horizontal line $y = 4$ , as shown in the diagram.

Find the coordinates of the point where the curve $x = \sqrt{4-y}$ intersects the $y$ -axis.

[2]

Find the area of the shaded region enclosed by the curve, the $y$ -axis, and the horizontal line $y = 4$ .

[4]

The region bounded by the curve is now rotated about the $y-axis$ , find the volume of revolution.

[4]

Question 3

HLPaper 2

The function $g(x) = x^2$ is defined over the interval $0 \leq x \leq 3$ .

Find the area of the shades section

[3]

Rotate the area found in part 1 around the x-axis. Use integration to find the volume of the solid generated.

[3]

Now rotate the same area around the y-axis. Set up the integral for the volume of the solid but do not evaluate it.

[2]

Find the point(s) where the curve intersects the line ( y = 9 ).

[2]

Question 4

HLPaper 2

The function $f(x) = 4 - x^2$ is defined on the interval $0 \leq x \leq 2$ .

Find the area enclosed between the curve $f(x) = 4 - x^2$ and the x-axis over the interval $0 \leq x \leq 2$ .

[2]

Find the area of the region enclosed by the curve and the y-axis on the same interval.

[3]

Rotate the area found in part 1 around the x-axis. Use integration to find the volume of the solid generated.

[5]

Determine the coordinates of the points where the curve intersects the axes.

[2]

Explain how the volume would change if the area was rotated around the y-axis instead of the x-axis.

[1]

Question 5

HLPaper 2

At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow. The path of the ball is modelled by the equation $\binom{x}{y} = \binom{5}{0} + t \binom{u_x}{u_y - 5t}$ where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacement from the ground, both measured in metres, and $t$ is the time, in seconds, since the ball was launched.

$u_x$ is the horizontal component of the initial velocity
$u_y$ is the vertical component of the initial velocity. In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that $u_x = 8$ and $u_y = 10$

Find the initial speed of the ball.

[2]

Find the angle of elevation of the ball as it is launched.

[2]

Find the maximum height reached by the ball.

[3]

Assuming that the ground is horizontal and the ball is not hit by the arrow, find the $x$ coordinate of the point where the ball lands.

[3]

For the path of the ball, find an expression for $y$ in terms of $x$ .

[3]

An archer releases an arrow from the point (0, 2). The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed 60 m s⁻¹ and an angle of elevation of 10°. Determine the two positions where the path of the arrow intersects the path of the ball.

[4]

Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.

[4]

Question 6

HLPaper 2

A curve is defined by the function $y = 9 - x^2$

Find the points where the curve intersects the (x)-axis.

[3]

Set up an integral to calculate the area between the curve and the (x)-axis from (x = -3) to (x = 3).

[4]

Evaluate the integral found in part 2.

[5]

Rotate the region enclosed by the curve between (x = -3) and (x = 3) about the (x)-axis. Set up the integral to find the volume of the solid generated.

[6]

Question 7

HLPaper 2

Daniel wants to build an open rectangular box to store items. The base and height of the box are both $(x)$ meters, and the length of the box is $(y)$ meters. The total volume of the box must be 36 m³. Your task is to determine the dimensions of the box that minimize the surface area of the materials used, and calculate the cost of painting the outside surface.

The outside surface area of the box is given by the formula:

$A(x) = \frac{72}{x} + 2x^2$

where $(x)$ is the side length of the base and height. The cost of painting is $20 per tin, with each tin covering 10 m².

Show that the formula for the surface area is given by:

$A(x) = \frac{72}{x} + 2x^2$

[4]

Find the derivative $(A'(x))$ .

$A(x) = \frac{1}{2}x^2 + 3x + 1$

[2]

Find the value of $x$ that minimizes the surface area.

[3]

Find the total cost for the minimized surface area.

[4]

Question 8

HLPaper 2

The curve is given by $y = 4 - x^2$ , which intersects the $x$ -axis at two points.

Find the points where the curve intersects the $x$ -axis.

[3]

Set up and evaluate the integral to find the area enclosed between the curve and the $x$ -axis.

[5]

Rotate the region enclosed by the curve about the $x$ -axis. Write down the integral for the volume of the solid generated. Do not evaluate the integral.

[4]

Question 9

HLPaper 2

Consider a function $f (x)$ , for $x \geq 0$ .The derivative of $f$ is given by $f' (x) = \frac{6 x}{x^{2} + 4}$ .

The graph of $f$ is concave-down when $x > n$ .

Show that $f'' (x) = \frac{24 - 6 x^{2}}{{(x^{2} + 4)}^{2}}$ .

[4]

Find the least value of $n$ .

[2]

Find $\int \frac{6 x}{x^{2} + 4} d x$ .

[3]

Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = 1$ and $x = 3$ .The area of $R$ is $19.6$ , correct to three significant figures.

Find $f (x)$ .

[7]

Question 10

HLPaper 1

The production of oil $(P)$ , in barrels per day, from an oil field satisfies the differential equation $\frac{d P}{d t} = \frac{1000}{2 + t}$ where $t$ is measured in days from the start of production.

The production of oil at $t = 0$ is $20, 000$ barrels per day.

Find $\int_{0}^{5} \frac{1000}{2 + t} d t$ .

[1]

State in context what this value represents.

[1]

Find an expression for $P$ in terms of $t$ .

[4]

Determine $\int_{0}^{365} P (t) d t$ and state what it represents.

[2]

AHL 5.12—Areas under a curve onto x or y axis. Volumes of revolution about x and y Questionbank