At university we are given a voluntary hand in in the use of maple/matlab, in that regard I have a double integral I am in dire need to compute, using first cartesian then polarcoordinates.
http://www.math.ntnu.no/emner/MA1103/2012v/maplematlab2.pdf
I need to find
$ \iint_R f(x,y)\,\mathrm{d}A $
Where R is the region below the curve $y = 3 + \sqrt{9-x^2} \, |x|<3$ and above the line $y = x/3 + 2$, and $ \displaystyle f(x,y) = \frac{x}{\sqrt{x^2+y^2\,}\,\,}$.
After making a nice drawing, i thought the integral could be expressed like this
$ \int_{-3}^{3} \int_{x/3 + 2}^{3 + \sqrt{9-x^2}} f(x,y)\,\mathrm{d}y\,\mathrm{d}x \approx -2.01$
Is this correct? Now I also need to compute this double integral using polarcoordinates Firstly I thought the circle could be express as
$y = 3 + \sqrt{9 - x^2} \ , \ |x| \quad \Leftrightarrow \quad r_2 = 6 \sin \theta \ , \, \pi/4 < \theta < 3\pi/4$
and the line could be expressed as
$y = x/3 + 2 \quad \Leftrightarrow \quad r_1 = \frac{6}{3 \sin \theta - \cos \theta}$
Which finally give me the following integral in polar coordinates:
$\int_{\pi/4}^{3\pi/4} \int_{r_1}^{r_2} f(x,y) r \, \mathrm{d}r \, \mathrm{d}\theta \, \approx \, 17.28$
But this is wrong, I think it is due to my angles,how can I compute the double integral in a propper way using both cartesian and polar coordinates? I appreaciate all help =)