Taking it further...

The graph of the function \(y=f(x)\) between \(x=a\) and \(x=b\) is rotated by \(2\pi\) about the \(x\)-axis to obtain a solid of revolution. (Assume that \(f(x)\ge0\) for \(a\le x\le b\).) Obtain a formula for the (curved) surface area of this solid of revolution (that is, excluding the two circular ends).

(You may find the ideas in Cones useful here.)

What would the corresponding result be if the curve were described parametrically by \((x(t),y(t))\), and we wished to find the surface area of the solid of revolution obtained by rotating the curve between \(t=a\) and \(t=b\) by \(2\pi\) about the \(x\)-axis.

Now (re)visit Cutting spheres to answer the question there in light of what you have now learnt.

How would your answers to the first two questions above change if we instead rotated the curve about the \(y\)-axis?