Let R be the region enclosed by the x-axis, the graph y = x2, and the line x = 4. Write an integral expression for the volume of the solid whose base is R
whose slices perpendicular to the x-axis are semi-circles.
The region R is the same as in the example:
This solid is like the one in the example except that instead of the slices being squares, they're semicircles. The slice at position x is a semi-circle with diameter
y = x2 and thickness Δ x.
This solid really looks like a loaf of French bread. Since the diameter of a slice is x2 the radius is . The volume of a slice is the area of the semi-circle (half the area of a circle) multiplied by the thickness, or
The variable x goes from 0 to 4 in this region, so when we take the integral we get