Question

I'm confused where I went wrong.

On the lecture here from 9:10 to 11:10 you can see how the answer of Pi/8 is derived: http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/3.-double-integrals-and-line-integrals-in-the-plane/part-a-double-integrals/session-50-double-integrals-in-polar-coordinates/

Yet on Wolfram alpha we get Pi/3 if we use the x = cos(theta) * r, y= cos(theta) * r, which he states in the lecture is the same as using the r^2

Answer 1

http://www.wolframalpha.com/input/?i=Integrate%5B1+-+%28r*Sin%5Btheta%5D%29%5E2+-%28r*Cos%5Btheta%5D%29%5E2+%2C+%7Btheta%2C+0%2C+Pi%2F2%7D%2C+%7Br%2C+0%2C+1%7D%5D

Answer 2

@ganeshie8

Answer 3

Easy, you forgot to multiply the "area scale factor" when you move from cartesian to polar : \(r\).

http://www.wolframalpha.com/input/?i=Integrate%5Br*%281+-+%28r*Sin%5Btheta%5D%29%5E2+-%28r*Cos%5Btheta%5D%29%5E2%29+%2C+%7Btheta%2C+0%2C+Pi%2F2%7D%2C+%7Br%2C+0%2C+1%7D%5D

Answer 4

Oh wow! Indeed I did, thank you.

Answer 5

np

\(dx~dy = \color{red}{r}~drd~\theta\)

\( \color{red}{r}\) is called Jacobian; i think he's gonna cover its derivation in "change of variables" lecture...