Find the volume $V$ of the described solid $S$.
The base of $S$ is the triangular region with vertices $(0, 0), (4,0)$, and $(0, 4)$.Cross-sections perpendicular to the x−axis are squares.
My answer is $V = 2π\left(\frac{27}{2} - \frac{27}{5}\right) = \left(\frac{π}{5} \right)(135 - 54) = \frac{81π}{5} $
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$\begingroup$According to your description of the solid, the base of the square in the plane perpendicular to the $x$-axis goes from the $x$-axis to the line $y=4-x$, so the side length is $4-x$. The area of that square is, of course, $y^2$, and the values of $x$ for those perpendicular planes run from $0$ to $4$.
Therefore the volume you want is
$$\int_0^4 (4-x)^2\,dx$$
You can finish from here.
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