Where does \pi y^2\delta x come from?

The volume of a thin disc of radius y and thickness \delta x.

What must you do before integrating for a rotation about the y-axis?

Write x in terms of y and use y-limits.

Volume of the solid between an outer curve and an inner curve?

\pi\int\left(y{outer}^2-y{inner}^2\right)dx

Why is \pi\int(y1-y2)^2\,dx wrong?

The volumes of the discs are subtracted, not the radii.

Volume of a cone from y=\frac rhx?

\pi\int0^h\frac{r^2x^2}{h^2}\,dx=\frac13\pi r^2h

Volume of a sphere from y=\sqrt{r^2-x^2}?

\pi\int{-r}^{r}\left(r^2-x^2\right)dx=\frac43\pi r^3

Parametric volume about the x-axis (A2)?

V=\pi\int{t1}^{t2}y^2\frac{dx}{dt}\,dt

What limits do you use for a parametric volume (A2)?

The values of t at the ends of the curve, not x-values.

Volumes of revolutionEdexcel A-Level Further Maths: Flashcards

Card 1 of 120 of 12 known

Question

Volume of revolution about the $x$-axis?

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

Volume of revolution about the $x$ -axis?: $V=\pi\int_a^by^2\,\mathrm{d}x$
Volume of revolution about the $y$ -axis?: $V=\pi\int_c^dx^2\,\mathrm{d}y$
Where does $\pi y^2\delta x$ come from?: The volume of a thin disc of radius $y$ and thickness $\delta x$ .
What must you do before integrating for a rotation about the $y$ -axis?: Write $x$ in terms of $y$ and use $y$ -limits.
Volume of the solid between an outer curve and an inner curve?: $\pi\int\left(y_{\text{outer}}^2-y_{\text{inner}}^2\right)\mathrm{d}x$
Why is $\pi\int(y_1-y_2)^2\,\mathrm{d}x$ wrong?: The volumes of the discs are subtracted, not the radii.
Volume of a cone from $y=\frac rhx$ ?: $\pi\int_0^h\frac{r^2x^2}{h^2}\,\mathrm{d}x=\frac13\pi r^2h$
Volume of a sphere from $y=\sqrt{r^2-x^2}$ ?: $\pi\int_{-r}^{r}\left(r^2-x^2\right)\mathrm{d}x=\frac43\pi r^3$
Identity used to integrate $\sin^2x$ ?: $\sin^2x=\frac12(1-\cos2x)$
Parametric volume about the $x$ -axis (A2)?: $V=\pi\int_{t_1}^{t_2}y^2\frac{\mathrm{d}x}{\mathrm{d}t}\,\mathrm{d}t$
Volume of $y=\sqrt x$ from $0$ to $4$ about the $x$ -axis?: $8\pi$
What limits do you use for a parametric volume (A2)?: The values of $t$ at the ends of the curve, not $x$ -values.

Exam questions on Volumes of revolution

The region $R$ is bounded by the curve $y=\sqrt x$ , the $x$ -axis and the line $x=4$ .
The solid formed in (a) is cut so that only the part with $1\leq x\leq4$ remains. Find the exact volume of this part.2 marks
The curve $C$ has equation $y=\frac1x$ for $x>0$ . All lengths are in centimetres.
A vase is modelled by rotating the part of $C$ between $x=2$ and $x=5$ through $2\pi$ radians about the $x$ -axis. Find the exact volume of the vase.2 marks
The finite region $R$ is bounded by the curve $y=x^2$ and the line $y=2x$ .
Find the exact volume generated when $R$ is rotated through $2\pi$ radians about the $x$ -axis.3 marks

See the full worksheet

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).