(865, #5) Find the surface area of the part of the hyperbolic paraboloid
z = y² - x² that lies between the cylinders x² + y² = 1 and x² + y² = 4..

Solution: The surface is a saddle with a hole

f := (x,y)-> y^2 - x^2:
plot3d([r*cos(t), r*sin(t), r^2*(sin(t)^2 - cos(t)^2)], r = 1..2, t = 0..2*Pi,
color = navy, grid = [6, 25]);:

The surface area is equal to (Pi/6)(17^3/2 - 5^3/2):

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

1+4*x^2+4*y^2

h := (x,y) -> sqrt( 1 + 4*(x^2 + y^2)):
lprint(int(int(h(r*cos(t),r*sin(t))*r, r = 1..2), t = 0..2*Pi));

17/6*Pi*17^(1/2)-5/6*Pi*5^(1/2)

by: gm