Find the surface area for the following:
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Find the surface area for the following:

[From: ] [author: ] [Date: 12-05-28] [Hit: ]
What is the area?Since dz/dr = -2r,Evaluated between limits of 1 and 2, (17^(3/2) - 5^(3/2)) * 2*pi/3 = 123.......
The portion of the paraboloid z=9-x^2-y^2 between the cylinders x^2+y^2=1 and x^2+y^2=4

and please show working out.

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I'll give you a hint. See that EVERYTHING here is symmetric around the origin, so a good start is to convert to polar coordinates.

In polar coordinates, the problem becomes to find the surface area of the paraboloid

z = 9 - r^2

subject to the constraint that

1 <= r <= 2

Be careful, you will still need to integrate here. What is the area? You must integrate an area element that looks like

sqrt(1 + (dz/dr)^2) * 2*pi*r dr

Since dz/dr = -2r, we need to do the integration

int(sqrt(1 + 4*r^2) *2*pi*r dr) = (4*r^2 + 1)^(3/2) *2*pi/3

Evaluated between limits of 1 and 2, the definite integral is

(17^(3/2) - 5^(3/2)) * 2*pi/3 = 123.38595878857
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