Find the volume

Question

The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x². Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?

Answer 1

These curves intersect at the origin and (1,1). The vertical distance (delta y) between them will be the diameter of each semicircle. The radius will be half of that. The area of ONE semicircle will be 1/2*pi*radius^2.

Diameter = x - x^2

Radius = 1/2(x-x^2)

Area = 1/2*pi*r^2

We will add up the areas of the semicircles from x=0 to x=1 by integrating as follows:

V(x)=integ[from 0, to 1] (pi/2)*[(1/2)(x - x^2)]^2 dx

Simplify carefully, integrate using the Power Rule, and apply the limits following the Fundamental Theorem of Calculus.

The solution is pi/240 cubic units.