Arc length and surface area of revolutionEdexcel International 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?
What is ds?
Surface area when y=f(x) is rotated about the x-axis?
Surface area for a parametric curve rotated about the x-axis?
Surface area for a parametric curve rotated about the y-axis?
What does 2\pi y represent?
Which curve forms are never set for this topic?
Length of y=\frac23x^{\frac32} from x=0 to x=3?
If x=3t^2, y=2t^3, what is \left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2?
Substitution for \int x^3\sqrt{1+9x^4}\,dx?
What must you change when you substitute in a definite integral?

Exam questions on Arc length and surface area of revolution

The curve $C$ has equation $y=\frac23x^{\frac32}$ for $x\ge0$ .
Find the exact length of the arc of $C$ from $x=3$ to $x=8$ .2 marks
A curve $C$ has parametric equations $x=3t^2$ , $y=2t^3$ for $t\ge0$ .
Find the length of the arc of $C$ from $t=0$ to $t=\sqrt3$ .2 marks
The curve $C$ has equation $y=x^3$ for $0\le x\le1$ . The arc of $C$ is rotated through $2\pi$ radians about the $x$ -axis to form a surface.
Show that the area of the surface is $S=2\pi\int_0^1x^3\sqrt{1+9x^4}\,dx$ .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).