cylindrical coordinates

**wik_chick88** · April 22nd 2009, 10:43 PM

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 $z = 1 - x^2 - y^2$ .

**redsoxfan325** · April 22nd 2009, 11:08 PM

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.

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