Evaluate the expression...

Question

Evaluate.

∫∫r cos(x + 2y) dA

where {(x, y) : 0 ≤ x ≤ π, 0 ≤ y ≤ π/2}

Answer 1

∫∫rcos(x + y)dxdy = r∫₀^π∫₀^π/2cos(x +2y)dydx = r/2∫/_∫0^πsin(x + 2y)₀^π/2dx = r/2∫₀^π(sin(x + π) - sinx)dx =

-r∫₀^πsinxdx = rcosx₀^π = r(cosπ - cos0) = - 2r