Question

the region bounded by the quarter circle (x^2) + (y^2) =1. Find the volume of the following solid.

The solid whose base is the region and whose cross-sections perpendicular to the x-axis are squares.

Answer 1

"The solid whose base is the region and whose cross-sections perpendicular to the x-axis are squares."
means that z=2y
but since y=2sqrt(1-x^2) (on the circle), so z=2sqrt(1-x^2)

For example, at x=0, z=1,
at x=1, z=0.


The volume of the solid is then
∫∫∫dx dy dz
where the limits of integration are
for z: 0 to 2sqrt(1-x^2)
for y: -sqrt(1-x^2) to sqrt(1-x^2)
for x: -1 to 1