Where does the formula V=\pi\int y^2\,dx come from?

Slice into discs of radius y and thickness \delta x with volume \pi y^2\,\delta x, then sum and let \delta x\to0.

Volume when the region lies between y=f(x) above and y=g(x) below, about the x-axis?

V=\pi\inta^b\left[f(x)^2-g(x)^2\right]dx

Common error with two curves?

Squaring the difference (f-g)^2 instead of subtracting the squares.

What does the mean value represent geometrically?

The height of a rectangle on [a,b] with the same area as the region under the curve.

Volume when y=2x, 0\le x\le3, is rotated about the x-axis?

36\pi (a cone of radius 6 and height 3).

What must you do before using V=\pi\int x^2\,dy?

Write x^2 in terms of y and use limits in y.

How do you find the x-value where f equals its mean value?

Solve f(x)=\bar y and keep only roots inside the interval.

Volumes of revolution and mean valueAQA A-Level Further Maths: Flashcards

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Volume of revolution about the $x$-axis from $x=a$ to $x=b$?

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Volume of revolution about the $x$ -axis from $x=a$ to $x=b$ ?: $V=\pi\int_a^b y^2\,dx$
Volume of revolution about the $y$ -axis from $y=c$ to $y=d$ ?: $V=\pi\int_c^d x^2\,dy$
Where does the formula $V=\pi\int y^2\,dx$ come from?: Slice into discs of radius $y$ and thickness $\delta x$ with volume $\pi y^2\,\delta x$ , then sum and let $\delta x\to0$ .
Volume when the region lies between $y=\mathrm{f}(x)$ above and $y=\mathrm{g}(x)$ below, about the $x$ -axis?: $V=\pi\int_a^b\left[\mathrm{f}(x)^2-\mathrm{g}(x)^2\right]dx$
Common error with two curves?: Squaring the difference $(\mathrm{f}-\mathrm{g})^2$ instead of subtracting the squares.
Mean value of $\mathrm{f}$ over $a\le x\le b$ ?: $\frac{1}{b-a}\int_a^b\mathrm{f}(x)\,dx$
What does the mean value represent geometrically?: The height of a rectangle on $[a,b]$ with the same area as the region under the curve.
Volume when $y=2x$ , $0\le x\le3$ , is rotated about the $x$ -axis?: $36\pi$ (a cone of radius 6 and height 3).
Volume when $y=x^2$ , $0\le x\le2$ , is rotated about the $x$ -axis?: $\frac{32\pi}{5}$
What must you do before using $V=\pi\int x^2\,dy$ ?: Write $x^2$ in terms of $y$ and use limits in $y$ .
Mean value of $x^2$ on $[0,3]$ ?: $\frac13\int_0^3x^2\,dx=3$
How do you find the $x$ -value where $\mathrm{f}$ equals its mean value?: Solve $\mathrm{f}(x)=\bar y$ and keep only roots inside the interval.

Exam questions on Volumes of revolution and mean value

The region $R$ is bounded by the curve $y=x^2$ , the $x$ -axis and the line $x=2$ . Lengths are in centimetres.
Only the part of $R$ with $1\le x\le2$ is rotated through $2\pi$ radians about the $x$ -axis. Find the exact volume of the solid formed.2 marks
The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=3x^2-2x$ for $0\le x\le2$ .
Find the mean value of $\mathrm{f}$ over the interval $1\le x\le2$ .2 marks
The region $R$ is bounded by the curve $y=\dfrac{1}{\sqrt{2x+1}}$ , the coordinate axes and the line $x=4$ .
Show that when $R$ is rotated through $2\pi$ radians about the $x$ -axis, the volume of the solid formed is $\pi\ln3$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).