Find the volume of the solid bounded by $ y=0 , z=1-x^2, z=x^2-1, y+z=1 $

Question

i need to know the first steps and strategies of solving this kind of problems

i know that i need to find the limits of x and y and then do the integral of a certain functions by dxdy

Answer 1

Sketching the solid, as suggested in the comments above, is the first thing you should do.

After that, a good way to get started on this particular problem is to sketch the curves

$\;\;\;z=1-x^2$ and $z=x^2-1$ in the $xz$-plane.

This region is the projection of the solid in the $xz$-plane, so you can use this to find your limits for $x$ and $z$.

Then (if you are using a double integral) you can use the equations $y=0$ and $y+z=1$

to find the $y$-values corresponding to any point $(x,z)$ in the projection in the $xz$-plane.

The difference of these $y$-values gives the length of the solid corresponding to each point $(x,z)$,

so it is the function to be integrated.

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This should give $\displaystyle V=\int_{-1}^1\int_{x^2-1}^{1-x^2}(1-z)\;dz\;dx$