Evaluate the following integrals by converting to polar coordinates:

∫∫ (x+y)dA, R={(x,y)|x^2+y^2≥4, x^2+y^2≤16, R x≥0, y≥0}

Please explain step by step

Question

Evaluate the following integrals by converting to polar coordinates:

∫∫ (x+y)dA, R={(x,y)|x^2+y^2≥4, x^2+y^2≤16,                      R                                                               x≥0, y≥0} 

Please explain step by step

turingtest · Answer

@Janex are you there to discuss this?

turingtest · Answer

Graph these equations on the xy-plane.
You will find it to be the graph of two concentric circles. You will be integrating between their radii.
(notice that the smaller circle has the inequality $r\ge$, while the larger circle has $r\le$ )
Their polar formulas should become obvious at that point, as well as the bounds on $r$ and $\theta$
Using the same substitutions we did for the previous problem$$x=r\cos\theta$$$$y=r\sin\theta$$$$dA=rdrd\theta$$you can convert the integrand