1. Double Integration

Evaluate the following double integrals over the given regions:

$\displaystyle\int\displaystyle\int_R \dfrac{x}{\sqrt{x^2 + y^2}}$ dA

$R= {(x,y) | 0 < x < \sqrt{4y -y^2}, 0 < y < 2}$

I have a feeling that this is not easily intergrated, im not too sure how i would go about doing it.. help would be appreciated.

Many thanks

2. $\int\frac{x}{\sqrt{x^{2}+y^{2}}}dx=\sqrt{x^{2}+y^{ 2}}$

$|_{0}^{\sqrt{4y-y^{2}}}\frac{x}{\sqrt{x^{2}+y^{2}}}=2\sqrt{y}-y$

$\int[2\sqrt{y}-y]dy=\frac{4y^{\frac{3}{2}}}{3}-\frac{y^{2}}{2}$

Using the limits 0 to 2 we get $\frac{8\sqrt{2}}{3}-2$

3. I dont understand the first integration...

4. Originally Posted by brd_7
I dont understand the first integration...
To find $\int\frac{x}{\sqrt{x^{2}+y^{2}}} \, dx$ make the substitution $w = x^2 + y^2$.

Remember that y is treated as a constant in this integral and so $\frac{dw}{dx} = 2x \Rightarrow dx = \frac{dw}{2x}$.

Then the integral becomes $\int\frac{x}{\sqrt{w}} \, \frac{dw}{2x} = \frac{1}{2} \int\frac{1}{\sqrt{w}} \, dw = \frac{1}{2} \int w^{-1/2} \, dw = w^{1/2} = \sqrt{w}$.

Now substitute back that $w = x^2 + y^2$ and you get $\sqrt{x^{2}+y^{2}}$. Capisce?

5. Capisce

6. Anyway, this is just chain rule.

You can modify the integrand such that you can find the hidden derivative, the rest follows.