Discussion Overview
The discussion revolves around evaluating the double integral \( II_{5a} \) over the region \( R = [0,2] \times [-1,1] \) for the function \( xy\sqrt{x^2+y^2} \). Participants explore the implications of the function's symmetry and the limits of integration.
Discussion Character
- Mathematical reasoning, Debate/contested
Main Points Raised
- One participant proposes evaluating the double integral \( II_{5a} = \iint\limits_{R} xy\sqrt{x^2+y^2}\, dA \) over the specified region.
- Another participant notes that the function \( f(x,y) = xy\sqrt{x^2+y^2} \) is odd in \( y \) and concludes that the integral over \( y \) from -1 to 1 results in zero, suggesting \( II_{5a} = 0 \).
- Some participants express concern about a potential typo in the limits of integration, specifically questioning the upper limit of \( x \) being 2 instead of 1.
- One participant argues that regardless of whether the limit is 1 or 2, the \( y \)-integral remains zero, leading to the conclusion that the double integral is still zero.
Areas of Agreement / Disagreement
There is disagreement regarding the limits of integration, with some participants questioning a potential typo. However, there is a consensus that the double integral evaluates to zero based on the symmetry of the function.
Contextual Notes
The discussion includes uncertainty about the correct limits of integration and the implications of the function's symmetry on the evaluation of the integral.