(864, #3) Find the area of the surface of the part of the cylinder y² + z² = 9 that lies above the rectangle with vertices (0,0) , (4,0) , (0,2) , and (4,2).

Solution: The surface is in red, the domain of integration in yellow:

The surface area is 12 arcsin(2/3):

f := (x,y) -> sqrt(9-y^2):
g := (x,y) -> 1 + diff(f(x,y),x)^2 + diff(f(x,y),y)^2:
lprint( simplify( g(x,y) ) );

-9/(-9+y^2)

h := (x,y) -> 3/sqrt(9-y^2):
lprint(int(int(h(x,y), x = 0..4), y = 0..2));

12*arcsin(2/3)

by: gm