The base of S is the region enclosed by the parabola $y = 9 − 9x^2$ and the X - axis. Cross-sections perpendicular to the X - axis are isosceles triangles with height equal to the base.
2026-04-03 23:21:04.1775258464
Finding volume of enclosed region
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You want to take the volume as $$V=\int_a^bA(x)\, dx,$$
where $A(x)$ is a typical area slice. I think from comments you can see that $$A(x)=\frac{1}{2}BH,$$
where $B$ is the base of the triangle and $H$ is the height. Thus $$A(x)=\frac{1}{2}\left(9-9x^2\right)\left(9-9x^2\right).$$
It helps to sketch these things to see what is going on. Learn how to sketch 3 dimensional solids and your life will be much more enjoyable. Next you want the limits of integration. They are the the zeroes of the function $y$, this being where the function meets the $x$ axis, and trivially found. See if $$V=\int_{-1}^{1}\frac{1}{2}\left(9-9x^2\right)^2\, dx$$ meets with your mathematical sensibilities.