Volumes, Averages and Other Uses of the Integral: Quick Revision

A brief note on Volumes, Averages and Other Uses of the Integral. Read it before a test. Try each question at the end before you open its answer.

Where this sits. The Indian syllabus (NCERT Class 12 and JEE Main) asks for areas only. Volumes of revolution, the mean value of a function and the length of a curve belong to A-Level Further Maths, IB Higher Level and AP Calculus. Check your own syllabus for the exact list.

Volumes of revolution

• About the $x$-axis, discs: $V = \pi\displaystyle\int_a^b y^2\,dx$.
• About the $y$-axis, discs: $V = \pi\displaystyle\int_c^d x^2\,dy$. Write $x^2$ in terms of $y$ first.
• Washers: $V = \pi\displaystyle\int (R^2 - r^2)\,dx$. Square each radius, then subtract.
• Checks: a cone has $\dfrac{1}{3}\pi r^2 h$ and a sphere has $\dfrac{4}{3}\pi R^3$.

Mean value and accumulation

• Mean value: $\dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx$, the height of the rectangle with the same area.
• Reached at a point: a continuous $f$ equals its mean value at some $c$ in $[a, b]$.
• Total from a rate: $\displaystyle\int_a^b r(t)\,dt$. The unit is the unit of the rate times the unit of time.
• Displacement: $\displaystyle\int v\,dt$. Distance: $\displaystyle\int |v|\,dt$, split at the zeros of $v$.

Length of a curve

• Formula: $L = \displaystyle\int_a^b \sqrt{1 + [f'(x)]^2}\,dx$, from $ds^2 = dx^2 + dy^2$.
• Neat answers: they come from curves where $1 + [f'(x)]^2$ is a perfect square, such as $y = \cosh x$.
• Estimate: the sum of tiny chords is a little below the true length.

A worked example

The line $y = \dfrac{x}{3}$ for $0 \le x \le 9$ is turned about the $x$-axis. Find the volume.

1. Set up. The radius is $y = \dfrac{x}{3}$, so $V = \pi\displaystyle\int_0^9 \dfrac{x^2}{9}\,dx$.
2. Integrate. $\dfrac{1}{9} \times \dfrac{9^3}{3} = 27$.
3. Answer. $V = 27\pi$. This is a cone with $r = 3$ and $h = 9$, and $\dfrac{1}{3}\pi \times 9 \times 9 = 27\pi$ agrees.

Practice with new numbers

A program to try

The program checks three results of the note: the volume of a cone by thin discs, the mean value of $\sin^2 x$, and the length of a straight line.

Mistakes to avoid

• Squaring the difference of radii. A washer has area $\pi(R^2 - r^2)$, not $\pi(R - r)^2$.
• Forgetting to divide by $b - a$. Without it you have the total, not the mean value.
• Using displacement for distance. Split at the zeros of $v$ and add the sizes of the pieces.
• Leaving out the square root in a length. The length element is $\sqrt{1 + [f'(x)]^2}\,dx$.

Check yourself

1. Turn the line $y = 2x$ for $0 \le x \le 1$ about the $x$-axis. Find the volume.

2. Find the mean value of $\cos x$ on $\left[0, \dfrac{\pi}{2}\right]$.

3. Water flows in at $r(t) = 10 - 2t$ litres per minute for $0 \le t \le 5$ minutes. How many litres arrive?

4. A cart has $v(t) = t - 2$ metres per second for $0 \le t \le 4$ seconds. Find the displacement and the distance.

5. Find the length of $y = \dfrac{4}{3}x$ from $x = 0$ to $x = 6$.

Sources

• OpenStax, Calculus Volume 1, Chapter 6, sections 6.2 and 6.4.
• OpenStax, Calculus Volume 1, Chapter 5, sections 5.3 and 5.4.