Approximating surface area of hemisphere

[serverxeon]

Homework Statement

i am trying to apprixmate the surface area of a hemisphere.
i am approximating by cutting the sphere into cylinders of different radius, and using their curved surface area to approximate.

each cylinder will have a height of r.cos.theta and radius of r.sin.theta.

so the surface area should be =
integral [pi/2 to 0] (2.pi * r.sin.theta * r.cos.theta * d.theta)??

but that gives me -pi r^2 which is wrong...

anyidea where i went wrong?

 

[HallsofIvy]

That's not going to work. Those cylnders do NOT approximate the surface area. The problem is exactly the same as if you used short vertical and horizontal segments to approximate the line from (0, 0) to (1, 1). They will always have a total length of 2 while the length of the the line is $\sqrt{2}$.

 

[serverxeon]

sorry i don't get you. can you elaborate?
how else can i integrate something to get the S.A.?
(without going through the volume method)