Question

Calculate the double integral ∫∫R xcos(2x+y)dA where R is the region: 0≤x≤π3,0≤y≤π²

Answer 1

To calculate the double integral of xcos(2x+y) over the given region, integrate with respect to y first, followed by x, and evaluate between the given limits.

Step-by-step explanation:

We are to calculate the double integral of the function xcos(2x+y) over the region R defined by the intervals 0≤x≤π/3 and 0≤y≤π². To solve the integral, we integrate with respect to y first and then with respect to x.

Steps to Solve the Double Integral

1. Firstly, integrate the function with respect to y, keeping x as a constant:
2. ∫ π² (xcos(2x+y)) dy = x[∫y cos(2x+y)] from 0 to π².
3. After integrating, evaluate the expression at the bounds.
4. Secondly, integrate the result with respect to x within the interval from 0 to π/3.
5. Finally, combine the results of the two integrations to find the total area under the curve for the given region R.