1. ## cylindrical coordinates

Use cylindrical coordinates to evaluate
$\displaystyle \int\int\int_{E} (x^3 + xy^2) dV$, where $\displaystyle E$ is the solid in the first octant (i.e. $\displaystyle x, y, z \geq 0$ that lies beneath the paraboloid $\displaystyle z = 1 - x^2 - y^2$.

2. Originally Posted by wik_chick88
Use cylindrical coordinates to evaluate
$\displaystyle \int\int\int_{E} (x^3 + xy^2) dV$, where $\displaystyle E$ is the solid in the first octant (i.e. $\displaystyle x, y, z \geq 0$ that lies beneath the paraboloid $\displaystyle z = 1 - x^2 - y^2$.
$\displaystyle x = \cos\theta$
$\displaystyle y = \sin\theta$
$\displaystyle z = z$
$\displaystyle r^2=x^2+y^2$
$\displaystyle dV=r\,dz\,dr\,d\theta$

So the bounds are:
$\displaystyle \theta: 0\to\pi/2$
$\displaystyle r: 0\to 1$
$\displaystyle z: 0\to 1-r^2$

So, $\displaystyle (x^3 + xy^2)\,dV = x(x^2+y^2)\,dV = r^2\cos(\theta)r\,dz\,dr\,d\theta$

Your integral is $\displaystyle \int_0^{\pi/2}\int_0^1\int_0^{1-r^2}r^3\cos(\theta)\,dz\,dr\,d\theta$

I'll let you integrate it.