find the area of the surface the part of the sphere X^ 2 + Y^ 2 + Z^ 2 = a^ 2 th

Question

find the area of the surface the part of the sphere X^ 2 + Y^ 2 + Z^ 2 = a^ 2 that lies within the cylinder x^2 + y^2 = ax and above the xy plane

Explanation / Answer

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.

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