Help with volume please!

Question

The base of a solid in the region bounded by the parabola x² + y = 4 and the line x + y = 2. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?

Options:

1. 
2. 
3. 
4. 

Answer 1

I plot the parabola and the line in the x,y plane to find an enclosed figure going from x = -1 to x = 2. For semicircles as described, the area at each x is π r² /2, where r is 1/2 the distance in y between the two curves: r = [(-x² +4) - (-x +2)] /2. To find r² we simplify the contents of the preceding [] and multiply that by itself, then divide by 4.

Each incremental volume element dV = π r² dx /2. Then V = π/8 ∫_-1² (x⁴ -2x³ -3x² +4x +4) dx = π/8 [x⁵/5 -1/2 x⁴ - x³ +2x² +4x] evaluated between -1 and 2. This boils down to π/8 (32/5 + 1/5 +1/2 +1) = 81π/80, answer #3!