Let’s look at #35 on p. 445. First we’ll plot the bounding curves and the axis of rotation.

f[x_]:=(Sin[x])^2
g[x_]:=(Sin[x])^4

Plot[{f[x],g[x]},{x,0,Pi}]

Here we can use the shell method since the equations are already solved for y. There are two integrals because f[x] is bigger than g[x] on the right half of the interval, and vice versa on the left half.

Integrate[2*Pi*(x+1)*(f[x]-g[x]),{x,Pi/2,Pi}]+Integrate[2*Pi*(x+1)*(g[x]-f[x]),{x,0,Pi/2}]

The next command just simplifies the prior output (i.e., ‘%’ refers to the most recently produced output.... that is not necessarily the one that is directly above; it evaluates the output from the last command you executed).

Simplify[%]