Double Integral in Polar Coordinates

[Easy_as_Pi]

Homework Statement

Evaluate $\int\int$_D(x+2y)dA, where D is the region bounded by the parabolas y=2x² and y=1+x²

Homework Equations

dA = r*drd$\vartheta$
r²=x²+y²

The Attempt at a Solution

Well, I know I need to put D into polar coordinates, but I'm lost on this one. The examples I have seen before involve D being a region like x²+y² = 16, which I could easily turn into 0$\leq$r$\leq$4 and 0$\leq$$\vartheta$ $\leq$2$\pi$

So, any advice on getting better at setting up these regions in terms of polar coordinates?
EDIT: Is r from 0 to 2 and theta from 0 to pi?

 

[anonymity]

Did your book ask you (explicitly) to use polar coordinates? If not, I do not think that it makes very much sense to convert to polar.

 

[Easy_as_Pi]

It's not from my calculus book, rather a homework sheet given by my professor. It does not explicitly state to solve with the polar coordinate system, but, as it's the unit we have been studying in class, I assumed it was how our professor wanted us to solve the homework problems. The next problem involves finding the volume of the paraboloid z=x^2 +y^2 above the region D in the xy-plane bounded by the line y=x-1 and y^2= 2x+6. Both of these seem much easier to solve using the cartesian coordinates. Assuming I have to solve them with polar coordinates, how would I find my limits of integration?

 

[anonymity]

I'm taking multivariable calc right now as well, and I'm not sure how your book is designed, but we learned double integration in polar coordinates right after (ie the section after) we learned double integration over general regions (the method which i would personally use to solve this problem if given the option).

If your book is designed similarly, it does seem possible that you would have a multitude of different questions on the same worksheet.

However, under the assumption that you DO need to use polar coords, I don't think that it would be possible without using x = r*cos[t] and y = r*sin[t].

 

[Easy_as_Pi]

Hey, so, y'all are probably right. I solved both of them without polar coordinates, and they were rather easy problems. I just got tripped up, because our last test had double integrals without polar coordinates, so I assumed we were supposed to be using new methods to solve those two.

Now, as I do need help figuring out how to determine the bounds of integration in polar coordinates (those two problems obviously are not going to help with that) if my region, D, were represented by x^2+y^2 = 2x. How do I find r and theta? In this case, the circle is not centered at the origin, so, both r and theta must change from the standard 0 to 2pi for theta, and radius being the bounds for r. Would theta now be -pi/2 to pi/2?