Find the volume of the solid whose base is the semicircle y= sqrt(1− x^2) where −1≤x≤1, and the cross sections perpendicular to the x -axis are squares.

Question

Find the volume of the solid whose base is the semicircle  y= sqrt(1− x^2) where  −1≤x≤1, and the cross sections perpendicular to the  x -axis are squares.

Steve · Accepted Answer

the base of the square at x is y=√(1-x^2), so its area is 1-x^2. Then, adding up all these thin slices of thickness dx, and taking advantage of symmetry, v = 2∫[0,1] (1-x^2) dx