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The solid whose base is the region bounded by the curves $y=x^2$ and $y=2-x^2$ and whose cross sections through the solid perpendicular to the x-axis are squares

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    Given an $x$, what would the be the length of the side of the square at that point? (And you're going to get some flax for posting homework with no explanation or background. People are going to want to know what you attempted and where you got stuck.) – 2017-01-22

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The graphs of $y=x^2$ and $y=2-x^2$ intersect at $x=\pm 1$.

Therefore, we integrate the square slices from $-1$ to $1$.

We get the width of each square as $w = (2-x^2)-x^2 = 2-2x^2$.

Squaring this gives us $(2-2x^2)^2$, the area of each square.

The volume integral is $\int_{-1}^1 (2-2x^2)^2 \,dx$.

Evaluating this gives us the volume as $\frac{64}{15}$ units$^3$