# cylindrical coordinates

• Apr 22nd 2009, 10:43 PM
wik_chick88
cylindrical coordinates
Use cylindrical coordinates to evaluate
$\int\int\int_{E} (x^3 + xy^2) dV$, where $E$ is the solid in the first octant (i.e. $x, y, z \geq 0$ that lies beneath the paraboloid $z = 1 - x^2 - y^2$.
• Apr 22nd 2009, 11:08 PM
redsoxfan325
Quote:

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

$x = \cos\theta$
$y = \sin\theta$
$z = z$
$r^2=x^2+y^2$
$dV=r\,dz\,dr\,d\theta$

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

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

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

I'll let you integrate it.