Question

The base of a solid is a circle with radius 3. Each cross section perpendicular to a fixed diameter of the base is semicircular. Find the volume V of the solid.

V =

Answer 1

Consider that the xy-plane is the base of your solid.
Thus the slices are actually vertical to the xy-plane .. . .

Put the center of the circular base on the origin.

Thus the equation of the boundary of the solid is x^2 + y^2 = 36
Without any loss of generality, make the x-axis the fixed diameter and thus the slices are vertical and perpendicular to the x-axis.


Note that your slice intersects the circular boundary in two points. And that is the new diameter of the circular slice.

Thus the radius of each slice is y.

V = ?(-6 , 6) A(x) dx
but A(x) is a semi-circle with radius y.. . .. .
A(x) = (p/2) y^2
A(x) = (p/2) (36 - x^2) .... this is the boundary of the base.

V = ?(-6, 6) (p/2) (36 - x^2) dx
now, you can solve that ... ...