The base of a solid enclosed between the graphs of y = sinx and y= -sinx from x=0 to x = pi . Each cross-section is perpendicular to the x-axis is a semicircle with the diameter connecting the two graphs . Find the volume of the solid. I know the definite integral goes from 0 to pi , a = 0, b= pi, but I'm unsure of what to do with y= sinx and y= -sinx .

Answer 1

the diameter of each semicircle is the distance between the two graphs:
sinx - (-sinx) = 2sinx
So, the radius of each disc is just sinx.

So, adding up the volumes of all the small semi-circular discs of thickness dx, you get
∫[0,π] 1/2 π sin^2(x) dx = π^2/4