Double integral limits after mapping

[nickthegreek]

I have the double integral,

∫∫sqrt(x^2+y^2) dxdy, and the area D:((x,y);(x^2+y^2)≤ x)

By completing the squares in D we get that D is a circle with origo at (1/2,0), and radius 1/2. Then I tried changing the variables to x=r cosθ+1/2, y=r sinθ and J(r,θ)=r which leads to a not so nice integrand, and stop.

By changing to polar coordinates directly we get that D transforms into E:((r,θ);r^2≤ r cosθ)) which obv equals r≤cosθ, and the integrand r^3, which is nice. Now to my question. What do I know of "E"? What would it look like? What´s the limits?

Ps. How do I write Latex here?
edit, Ooops, wrong part of the forum? Sry, ill post in the textbook-style problem-part instead...

 

[jvicens]

Hello,

Thank you for sharing your approach to solving this double integral problem. It seems like you have made some good progress by completing the squares and transforming the variables to polar coordinates. Let me address your questions:

1. What do we know about "E" and what would it look like?

"E" represents the transformed area in polar coordinates, which is the same as the original area D. In this case, "E" is a circular region with a radius of 1/2 and centered at (1/2,0). This can be visualized as a quarter of a circle, with the origin at the center and the radius extending from the origin to the outer edge of the circle.

2. What are the limits for "E"?

The limits for "E" would be r from 0 to 1/2 and θ from 0 to π/2. This is because the radius of the circle is 1/2 and the angle θ ranges from 0 to π/2 as we move from the positive x-axis to the positive y-axis.

3. How do I write Latex here?

To write Latex here, you can use the dollar sign ($) to enclose your Latex code. For example, to write x^2, you can use $x^2$ and it will appear as x^2 in your post.

I hope this helps clarify your questions. Good luck with your problem-solving!