(865, #9) Find the surface area of the part of the sphere x² + y² + z² = a² that lies within the cylinder x² + y² = ax and above the xy-plane.

Solution: First, by completing the square, one finds that the equation of the cylinder is x² + (y - a/2)² = (a/2)². Thus the cylinder is a circle of radius (a/2), centered at (0, a/2).

f := (x,y)-> sqrt(4-x^2 - y^2):

plot3d({[r*cos(t), 1+r*sin(t), f(r*cos(t), 1+r*sin(t))],[r*cos(t), 1+r*sin(t), 0]},
r = 0..1, t = 0..2*Pi, color = blue, grid = [6,25], labels = [` x`, ` y`,`z`],
labelfont = [HELVETICA, BOLD, 10], orientation = [28,77], axes = framed);

The surface is some type of sea-creature:

The polar equation of this circle is r = a cos(t), t = 0 ..Pi; this is used in setting up the limits of integration below. The surface area is equal to a² (Pi - 2).

f := (x,y) -> sqrt(a^2 -x^2 - y^2):

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

a^2/(a^2-x^2-y^2)

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

Pi*(a^2)^(1/2)*a-4^(1/2)*(a^2)^(1/2)*a

This last line is equal to a²(Pi - 2). Maple is very cautious about taking square roots.

by: gm