(937, #21) 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 area is given by the integral

where A is the ring-shaped region formed by the circles x² + y² = 1 and x² + y² = 4.

The integral can be evaluated by switching to cylindrical coordinates:

To graph this with Maple use the parametric representation

plot3d([r*cos(t), r*sin(t), r^2 * (sin(t)^2 - cos(t)^2)], r = 2..4, t = 0..2*Pi,
grid = [10,25], color = blue, tickmarks = [2,3,2], labels = [`x`, `y`, `z`],
labelfont = [HELVETICA, BOLD, 10]);