Find the volume of the solid whose base is the region bounded by = ² and the line = 1, and whose cross sections perpendicular to the base and parallel to the x-axis are squares.
I'm not sure what the area of the cross section would be. The cross section is a square so it would just be the side length ² but I don't know how to figure out what that side length is. I know that I need to find the area of the cross section and then I can find the volume by doing the integral of the area of cross section
The easiest way to think of this is to recognize that all slices of the volume are squares that are parallel to the $x-z$ plane. At any value of $y$, the chord will be $2x=2\sqrt{y}$. Thus the area of a vertical square is $A=4y$. Then the volume is simply
$$V=\int_0^1 A\ dy=4\int_{0}^{1}y\ dy=2$$