Help me with double integrals

**zpwnchen** · October 24th 2009, 08:35 AM

$\int_D \int (2x-y) dA$
D is bounded by the circle with center the origin and radius 2.

help me to get started.

**tonio** · October 24th 2009, 09:19 AM

Originally Posted by zpwnchen

help me to get started.

Well, as that circle's equation is $x^2+y^2=4 \Longrightarrow y=\pm\sqrt{4-x^2}$ , we can see that

$\iint\limits_D (2x-y)dA=\int\limits_{-2}^2\int \limits_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(2x-y)dydx$

Tonio

**Mush** · October 24th 2009, 09:20 AM

Originally Posted by zpwnchen

help me to get started.

When dealing with circles, I would suggest converting to polar coordinates.

Do you know the following formulae:

$x = r \cos (\theta)$

$y = r \sin (\theta)$

Also, do you know the Jacobian matrix for a change of variables in double integration?

Once you have converted to polar coordinates, your variables of integration will be r, the radius, and $\theta$ , the angle. Clearly, the limits for these are as follows:

$0 \leq r \leq 2$

$0 \leq \theta \leq 2\pi$

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