Having trouble finding the volume:

Solid whose base is the region bounded by y=x^2 and y = 9. And whose cross sections are semi circles perpendicular to the base and parallel with the x-axis.

I know the interval should be between 0 and 3, but what confuses me is the semi-circle cross section part.

Question

Having trouble finding the volume:

Solid whose base is the region bounded by y=x^2 and y = 9. And whose cross sections are semi circles perpendicular to the base and parallel with the x-axis.

I know the interval should be between 0 and 3, but what confuses me is the semi-circle cross section part.

anonymous · Answer

The volume will essentially be the sum of the cross-sections' areas. The area of each cross section is $\dfrac{1}{2}\pi r^2=\dfrac{1}{2}\pi\left(\dfrac{d}{2}\right)^2=\dfrac{1}{8}\pi d^2$, since they are all semicircles. Each cross-section is parallel to the x-axis, which means the diameter of each cross section should be expressed in terms of y:
$$V=\int_a^b A(y)~dy$$
Each cross-section's diameter is the distance from one end of the parabola to the other:
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