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Related rates and volumes of revolution

Apply the chain rule to problems involving related rates of change; calculate volumes of solids of revolution by integrating πy2\pi y^2 (rotation about the xx-axis) or πx2\pi x^2 (rotation about the yy-axis).

Worked examples

Related rates: expanding circle

Straightforward

Problem

Water ripples outward in a circular pattern. The area of the circle is increasing at 12 m2/s12 \text{ m}^2/\text{s}. At what rate is the radius increasing when the radius is 4 m?

Related rates: conical tank

Moderate

Problem

A conical tank (apex down) has height 8 m and base radius 4 m. Water flows in at 3 m3/min3 \text{ m}^3/\text{min}. How fast is the water level rising when the depth is 2 m?

Volume of revolution: between two curves

Challenging

Problem

Find the volume of the solid formed by rotating the region between y=xy = x and y=x2y = x^2 (for 0≤x≤10 \le x \le 1) about the xx-axis.

Practise

Q1·Straightforward
A spherical balloon is being inflated so its volume increases at 20 cm3/s20 \text{ cm}^3/\text{s}. Find the rate at which the radius is increasing (in cm/s) when the radius is 5 cm. Give your answer to 4 decimal places. (Use V=43πr3V = \tfrac{4}{3}\pi r^3.)
Q2·Straightforward
A 10 m ladder leans against a vertical wall. Its base slides outward along the floor at 2 m/s2 \text{ m/s}. Find the rate at which the top of the ladder is sliding down the wall (in m/s) when the base is 6 m from the wall. Give a positive value for the speed of descent.
Q3·Moderate
A circular oil spill expands so that its area increases at 5 m2/min5 \text{ m}^2/\text{min}. Find the rate at which the radius is increasing (in m/min) when the radius is 3 m. Give your answer to 4 decimal places.
Q4·Moderate
A conical tank with its apex at the bottom has a height of 6 m and a base radius of 3 m. Water drains out at 2 m3/min2 \text{ m}^3/\text{min}. How fast is the water level falling (in m/min) when the depth is 3 m? Give your answer to 4 decimal places.
Q5·Straightforward
Find the volume of the solid formed by rotating y=xy = \sqrt{x} about the xx-axis from x=0x = 0 to x=4x = 4. Give your answer to 2 decimal places.
Q6·Straightforward
Find the volume of the cone formed by rotating y=xy = x about the xx-axis from x=0x = 0 to x=5x = 5. Give your answer to 2 decimal places.
Q7·Moderate
Find the volume of the solid formed by rotating y=2x−x2y = 2x - x^2 about the xx-axis from x=0x = 0 to x=2x = 2. Give your answer to 2 decimal places.
Q8·Moderate
Find the volume of the solid formed by rotating y=exy = e^x about the xx-axis from x=0x = 0 to x=1x = 1. Give your answer to 2 decimal places.
Q9·Challenging
Find the volume of the solid formed by rotating the region between y=xy = \sqrt{x} and y=xy = x (for 0≤x≤10 \le x \le 1) about the xx-axis. Give your answer to 4 decimal places.
Q10·Moderate
Find the volume of the solid formed by rotating y=x2+1y = x^2 + 1 about the xx-axis from x=0x = 0 to x=2x = 2. Give your answer to 2 decimal places.
Q11·Challenging
Find the volume of the solid formed by rotating y=sin⁡xy = \sin x about the xx-axis from x=0x = 0 to x=πx = \pi. Give your answer to 4 decimal places.
Q12·Challenging
Use the volume of revolution formula to prove that the volume of a sphere of radius rr is V=43πr3V = \dfrac{4}{3}\pi r^3. Start by rotating the semicircle y=r2−x2y = \sqrt{r^2 - x^2} about the xx-axis from x=−rx = -r to x=rx = r.

✎ Work this one through on paper — proofs are self-assessed.