( 842, #17) Calculate the double integral

where R = { (x, y) : 1 <= x <= 4, 0 <= y <= Pi/6}.

Solution:

f := (x,y)-> x*sin(y):
c := x ->0: d := x -> Pi/6:
Graph_dydx(f, 1, 4, c,d, 10);

Integrate_dydx(f,1,4, c, d);

f(x,y) = x*sin(y) c(x) = 0 d(x) = 1/6*Pi

int(f(x,y), y) = -cos(y)*x

int(f(x,y), y = c(x) .. d(x)) = -1/2*3^(1/2)*x+x

int( int(f(x,y), y = c(x) ..d(x)), x) = -1/4*3^(1/2)*x^2+1/2*x^2

int(int(f(x,y), y = c(x) ..d(x)), x = a .. b) = -15/4*3^(1/2)+15/2

visual check: average value = (-15/4*3^(1/2)+15/2)/(4*Pi/6) = 0.4797611820

by: rjm