Can somebody explain how to find the volume of a region enclosed between a surface and various planes.
For example -
If $S$ is a region enclosed by the surface $z= y^2$ and the planes $z=1$ , $x=0$ , $x=1$, $y=-1$ and $y=1$ what is the volume of $S$?
2026-02-22 21:51:06.1771797066
Volume of a region enclosed between a surface and various planes
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If you draw it (first the function $z=y^2$ and then extend the $x$-dimension), then you will see that you need to calculate the volume of a $1\times 2\times 1$-cuboid minus the volume under $z=f(x,y)=y^2$ on $[0,1]\times [-1,1]$, so just calculate $\int_0^1\int_{-1}^1 f(x,y)dydx$ and substract this from the volume of the cuboid to find the volume of $S$.