Question

Evaluate the given integral by changing to polar coordinates: ∫ ∫(Subscript D) e^((-x^2)-y^2) dA, where D is the region bounded by the semicircle x = √(9 - y^2) and the y-axis.

Answer 1

What problem are you having?

Answer 2

I have no idea where to even start...

Answer 3

There are three main steps. Change your integrand to polar, express dA in terms of polar coordinates and express your region in polar.

Answer 4

How do I change \[e^(-x^2-y^2)\] into polar coordinates?

Answer 5

x^2+y^2=r^2

Answer 6

So it would be \[e^(-r^2)\]?

Answer 7

Yes

Answer 8

Then what does dA stand for in polar coordinates?

Answer 9

Since you are changing to polar from cartesian. \[dxdy=\frac{\partial (x,y)}{\partial (r, \theta)}drd\theta\]

Answer 10

Does this mean \[\frac{ dx }{ dy } = \frac{ dr }{ d \theta }\] ?

Answer 11

I mean:

Answer 12

Have you not done change of variables in multivariable?

Answer 13

It was briefly covered

Answer 14

We know x=rcos(theta), y=rsin(theta). THe Jacobian of the transformation in this case is
\[\frac{\partial(x,y)}{\partial (r, \theta)} = \det \begin{bmatrix} \frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta}\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta} \end{bmatrix}\]

Answer 15

ah, thank you