( 842, #19) Calculate the double integral

where R = [0, Pi/6] x [0, Pi/3]

Solution:

f := (x,y)-> x*sin(x+y):
c := x -> 0: d := x -> Pi/3:
Graph_dydx(f, 0, Pi/6, c,d, 10);

The Maple output on integration is rather complicated. In addition, an integration by parts is required. So we present a standard argument:

Then, integrating by parts: u = x, du = dx, dv = sin(x + y)dx, v = -cos(x + y)

Some numbers: The value of the integral is 0.1042260162:

evalf(-1/12*Pi-1/2+1/2*3^(1/2));

Visual check: average = (0.1042206162)/(Pi^2/18) = .1900854599

by: nl