How to sketch this Domain of triple integral

Question

i am having hard time sketching the domain of this :

$$ \ \int_0^1\int_0^{1-x^2}\int_0^y f(x,y,z){dz}{dy}{dx} $$

is there an easy way to do that ? i got like cylinder and planes and its hard to see the Volume domain

The question asking to sketch this " simple " domain

Answer 1

You integrate with respect to $z$ first, keeping $x,y$ fixed. For any $(x,y), z$ ranges from $0\to y$. That means you're integrating below the plane $y=z$. Next you integrate with respect to $y$, keeping $x$ fixed. $y$ ranges from $0\to1-x^2$. This means you are integrating over the region below the parabola $y=1-x^2$, from $x=0\to1$, where it intersects the $x$ axis.

Answer 2

We have that

• in the $x$-$y$ plane the domain is the area between the parabola $y=1-x^2$ and positive axis
• coordinate $z$ varies between $z=0$ and $z=y$

therefore you can think to a cylinder with base $\{(x,y):x,y>0 \land y\le 1-x^2\}$ cutted by the plane $z=y$.

Answer 3

is this sketch right ? its like a cylinder cutted by z = y