Find the volume of an object using integrals

Question

How am I supposed to find the volume of an object when I know that: $$x^2+y^2\le z^2, \ 0 \leq x, \ 0 \leq y, \ 0 \leq z \leq 1$$

Answer 1

I recommend to look at the cut's of the plains that parallel to x-y plain. let's also neglect $x,y > 0$ constraint at that moment.

the integral you have to solve is: $$\iiint 1 \,dx\,dy\,dz$$

you can integrate the cut, as I previously said.

$$\iiint 1 \,dx\,dy\,dz = \int_{0}^{1}dz*\iint_{x^2+y^2 < z^2} dxdy$$ each cut, we have the area of disk with radius $z$.(because $x^2+y^2 < z^2$). the area of that disk is $\pi*z^2$,so all you have to do now is: $$\int_{0}^{1} \pi z^2 dz$$

so the answer is $\pi/3$
in your question you added that $x,y > 0$ so you have exactly $1/4$ disk at each cut so , divide $\pi/3$ by $4$ and you get $\pi/12$