Arc length and surface area of revolutionAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

Arc length of y=f(x) from x=a to x=b?
Arc length of a parametric curve?
Arc length of x=g(y) from y=c to y=d?
Surface area about the x-axis (Cartesian)?
Surface area about the x-axis (parametric)?
Surface area about the y-axis (Cartesian y=f(x))?
Why is there a factor 2\pi in a surface area of revolution?
What must the limits be in a parametric arc-length integral?
Why can \sqrt{36t^2(1+t^2)} be written 6t\sqrt{1+t^2}?
Which is volume and which is surface area: \pi\int y^2\,dx or 2\pi\int y\,ds?
Arc length of y=\frac23x^{3/2} from x=0 to x=3?
Surface area from rotating y=\sqrt x, 0\le x\le2, about the x-axis?

Exam questions on Arc length and surface area of revolution

The curve $C$ has equation $y=\frac23x^{3/2}$ for $0\le x\le3$ .
Find the exact length of $C$ .2 marks
The curve $C$ has equation $y=\sqrt x$ for $0\le x\le2$ . It is rotated through $2\pi$ radians about the $x$ -axis to form a surface.
Write down an integral, in terms of $x$ , for the area of the surface formed when $C$ is rotated through $2\pi$ about the $y$ -axis. Do not evaluate it.2 marks
A curve $C$ is given parametrically by $x=3t^2$ , $y=2t^3$ for $0\le t\le1$ .
Show that the length of $C$ is given by $\int_0^1 6t\sqrt{1+t^2}\,dt$ .3 marks

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).