Problems

01

Surface area: revolved cubic

The curve $y=x^3$ over $x\in [\mathrm{0,2}]$ is revolved around the $x$-axis.

Find the area of the resulting surface.

02

Surface area: cone

A cone may be described as the surface of revolution of a ray emanating from the origin, revolved around the $x$-axis.

Let $f(x)=mx$ for some $m>0$. Find the surface area of the cone given by revolving the graph of $f$ around the $x$-axis over $x\in [0,h]$.

Now calculate this area using geometry, and verify that the two methods give the same formula. (Hint: ‘unroll’ the cone into a sector.)

03

Surface area: parabolic reflector

A parabolic reflector is given by rotating the curve $y=x^2$ around the $y$-axis for $x\in [\mathrm{0,2}]$.

What is the surface area of this reflector?

04

Surface area: torus

A torus is created by revolving about the $x$-axis the circle with this equation:

$$x^2+{(y-b)}^2=a^2$$

Find the surface area of this torus.

(Hint: compute for the top and bottom of the circle separately and add the results.)