( 842, #9) Calculate the iterated integral

Solution:

f := (x,y)->sqrt(x+y):
c := y -> 0: d := y ->1:
Graph_dxdy(f,0,3,c,d, 9);

Integrate_dxdy(f,0,3,c,d);

f(x,y) = (x+y)^(1/2) c(y) = 0 d(y) = 1

int(f(x,y), x) = 2/3*(x+y)^(3/2)

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

int( int(f(x,y), x = c(y) ..d(y)), x) = 4/15*(1+y)^(5/2)-4/15*y^(5/2)

int(int(f(x,y), x = c(y) ..d(y)), y = a .. b) = 124/15-12/5*3^(1/2)

visual check: average = (4/15)(31 - 9 sqrt(3))/3 = 4.109744728/3= 1.369914909

I found this average hard to believe, so I inserted the plane z = 1.3699 level to check it out. This made it more plausible:

display3d({Graph_dxdy(f,0,3,c,d, 9),
plot3d(1.36991, x = 0..1, y = 0..3, color = green, grid = [3,8])} );

by: rjm