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

Plot[{f[x],0,-1},{x,0,Pi},PlotRange->All]

So we want to rotate the region bounded by the x-axis and the function f around the line y = - 1.  If you visualize a typical (vertical; perpendicular to the axis of rotation) rectangle in the region to be rotated, you see that we need to use the method of washers because there is space between the region being rotated and the axis of rotation.

The outer radius of the washer would be the distance between y = f[x] and y = -1, which is f[x] - (-1) or f[x]+1. The inner radius of the washer would be the distance between y = 0 and y = -1, which is 0 - (- 1) = 1. Thus we can use Mathematica to computer the following integral, which is the exact area of the solid of revolution.

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