Understanding what integrating in polar gives you

[tnmann10]

I am not understanding integration with polar coordinates, or at least visualizing what is happening. Here's the integral calculated in Wolfram:

http://www.wolframalpha.com/input/?i=integrate+(r^2(cost^2-sint^2))r+drdt+t=(0)..(pi/2)+r=(1)..(2)+

the integral before I changed it to polar was just ∫_R(x²-y²)dA where R is the first quadrant region between the circles of radius 1 and 2

mathematically it makes sense that the answer is 0.

but when you draw the picture it is a quarter of a washer in the xy plane. This does not seem like 0 to me. Would someone please explain where my thinking is going wrong?

Thanks

First post

 

[Office_Shredder]

The function x^2 -y^2 is symmetric in that region when you reflect over the line x=y, with the function negative on one half and positive on the other

 

[tnmann10]

ok that makes sense, but why the line x=y?

 

[chiro]

ok that makes sense, but why the line x=y?

Think about the nature of ^2 term in terms of the solutions (hint: + and -).