Convert an integral from Cartesian to Polar coordinates and evaluate it.

**asdf122345** · April 28th 2010, 05:22 PM

Need help with this.

**Amer** · April 29th 2010, 08:58 AM

Originally Posted by asdf122345

Need help with this.

Convert an integral from Cartesian to Polar coordinates and evaluate it.-graph223.jpg

$r^2 = x^2 +y^2$
$y = r\sin \theta$
$x = r \cos \theta$
lets convert the boundaries

$x = \sqrt{4y-y^2 }$

$x^2 = 4y -y^2 \Rightarrow x^2+y^2 = 4y$

$r^2 = 4r\sin \theta$

$r = 4 \sin \theta$

r changes from 0 to $4\sin \theta$

and theta from 0 to pi/2 the first quarter so

$\int_{0}^{4}\int_{0}^{\sqrt{4y-y^2}} x^2 \; dx dy$

becomes

$\int_{0}^{\frac{\pi}{2}} \int_{0}^{4\sin \theta} r^2 \cos^2 \theta (r) dr d\theta$

$\int_{0}^{\frac{\pi}{2}} \cos^2 \theta \left(\frac{r^4}{4}\right)\mid^{4 \sin \theta}_{0} d\theta$

$\int_{0}^{\frac{\pi}{2}} \cos^2\theta (4^3 \sin^4 \theta ) d\theta$

**asdf122345** · April 29th 2010, 03:12 PM

Is it possible to integrate where you left off? I can't figure out how to integrate that. Or does it get messy from integrating from that.

**Amer** · April 29th 2010, 10:07 PM

$4^3 \int \cos ^2 \theta (\sin ^4 \theta ) d\theta$

$\sin \theta \cos \theta = \frac{\sin 2\theta }{2}$

$\sin ^2 \theta = \frac{1-\cos 2\theta}{2}$

$4^3 \int \cos ^2 \theta \sin ^2 \theta ( \sin ^2\theta ) d \theta$

$4^3 \int \frac{\sin ^2 2\theta}{4} \left(\frac{1- \cos 2\theta }{2}\right) d\theta$

$4^3 \left(\int \frac{\sin ^2 2\theta}{4} d\theta - \int \frac{\sin ^2 2 \theta \cos 2\theta}{8} d\theta\right)$

first integral convert $\sin ^2 2\theta = \frac{1- \cos 4 \theta }{2}$

second one sub $u = \sin 2\theta$

can you continue ?

**asdf122345** · April 30th 2010, 10:46 AM

Yea I got it from here. Thanks

**asdf122345** · April 30th 2010, 01:13 PM

I got 4 pi as the answer. Is that correct? Something seems wrong.

**shenanigans87** · April 30th 2010, 01:19 PM

The answer is $2\pi$

So you may have just made a small error by a factor of 2.

Thread: Convert an integral from Cartesian to Polar coordinates and evaluate it.

LinkBack

Thread Tools

Display

Convert an integral from Cartesian to Polar coordinates and evaluate it.

Similar Math Help Forum Discussions

Change integral to polar coordinates and evaluate.

Transforming integral in cartesian coordinates to polar coordinates

convert polar coordinates to rectangular coordinates

[SOLVED] Convert polar coordinates to rectangular

convert it into polar coordinates,check limits

Search Tags