| Justaspirant:
Why did you take the integral of h(x) from 0 to 2 and then multiply it with the previously calculated area?
Visualize the region R as the base of a swimming pool. At x=0, it's deepest, and it gets shallower and shallower as x gets to 2, where the depth is 1.
At x=1, at every point y, the depth is 2. Imagine a vertical line going down x=1, down into the paper 2 feet deep. What you have is a rectangle with height 2 and side length equalling the distance of the vertical line. So what the pool really is is a bunch of rectangular slices from x=0 to x=2. This question is basically a hidden slice method question, with the width being the vertical distances along R and the height being 3-x.
The area of a rectangle is base times height. So the volume of the whole pool is the integral from 0 to 2 of [f(x)][3-x]dx, where f(x) is the sin(pi)x - (x^3-4x) |