Find the volume of the solid bounded by the surfaces $x = 1 − y^{2}$ , $x = −1$ and $z^{2} = 1 − x$

Question

Find the volume of the solid bounded by the surfaces

$x = 1 − y^{2}$ , $x = −1$ and $z^{2} = 1 − x$

I am having a bit of difficulty setting up the bounds for this question.

So far I have got:

• $-1 \leqslant x \leqslant 1-y^{2}$

• $-\sqrt{2} \leqslant y \leqslant \sqrt{2}$

• $-\sqrt{2} \leqslant z \leqslant \sqrt{2}$

Can someone please confirm if I am on the right track?

Answer 1

$$ V=\int_{-\sqrt 2}^{\sqrt 2} \int_{-1}^{1-y^2} \int_{-\sqrt {1-x}}^{\sqrt {1-x}} dz \;dx\;dy $$