# Thread: volume integral with 3 formulas mmn 15 3a

1. ## volume integral with 3 formulas mmn 15 3a

find the volume inclosed by z=0 x=y and y^2+z^2=x

i am having trouble drawing it
i know i should look on th shadow of it on the x-y plane and integrate by it

2. Originally Posted by transgalactic
find the volume inclosed by z=0 x=y and y^2+z^2=x

i am having trouble drawing it
i know i should look on th shadow of it on the x-y plane and integrate by it
No, you shouldn't. y^2+ z^2= x is a paraboloid with axis along the x-axis, not the y-axis. For this problem you should project onto the yz-plane.

Or, you can just "change" the problem- swap x and y and find the voume of the region bounded by x= 0, z= y, and z= x^2+ y^2. For that, you project onto the xy-plane.

3. i cant imagine it

4. I would approach it like this:
$\displaystyle \iint_D\int_{y^2+z^2}^{y}dxdydz$ where $\displaystyle D: y^2+z^2\leq y; z\geq0$

5. the main problem is how to know how y^2+ z^2= x looks like
i only know how y^2+ x^2= z

6. ## Re: volume integral with 3 formulas mmn 15 3a

Originally Posted by HallsofIvy
No, you shouldn't. y^2+ z^2= x is a paraboloid with axis along the x-axis, not the y-axis. For this problem you should project onto the yz-plane.

Or, you can just "change" the problem- swap x and y and find the voume of the region bounded by x= 0, z= y, and z= x^2+ y^2. For that, you project onto the xy-plane.
i can project on every plane i want to and build the integral appropriatly
i jast cant imagine this shape

if i can find out how the shape looks like
then it will solve it very fast
but imaganening a parabaloid cutting x=0 plane and z=0
is very hard thing to draw and imagine