- #1

realcomfy

- 12

- 0

The appropriate differential line element in cartesian coordinates is [tex] dx \hat{i} + dy \hat{j} [/tex] and after taking the dot product between the two I am left with

[tex]\int x^{2} dx + x y^{2} dy [/tex]

For the first term, I integrated x from -R to R (its max and min on the circle) while for the second term I set [tex] x = [tex]\sqrt{R^{2} - y^{2}}[/tex], leaving me with [tex] \int y^{2}* \sqrt{R^{2} - y^{2}} dy [/tex] and integrated over the same region. The first term was easy, and after some substitutions I was able to get the second term as well. However, I was left with 2*R^3/3 + Pi*R^4/8 and the answer was supposed to be simply Pi*R^4/4

So the second term is off by a factor of 1/2 and the first shouldn't even be there. Also, the second was only obtained by plugging in the values of an arctan term into Wolfram alpha, a method I am assuming I shouldn't have to use. If anyone can spot where I am making the error or if I am performing an operation that isn't allowed it would be greatly appreciated. Thanks for your time.