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Find the area of the surface given by z = f(x, y) over the region R. (Hint: The...

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ramiyaou | Student, Undergraduate | eNoter

Posted March 25, 2013 at 10:36 PM via web

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Find the area of the surface given by z = f(xy) over the region R. (Hint: The integral may be simpler in polar coordinates.)

f(xy) = 64 + x^2 − y^2
R = {(xy): x^2 + y^2 ≤ 1}

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mathsworkmusic | (Level 3) Associate Educator

Posted March 31, 2013 at 11:45 AM (Answer #1)

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We have the surface

`f(x,y) = 64 + x^2 - y^2`

and the constraint `x^2+y^2 <=1` , `x` and `y` `in R`

This constraint can be rewritten as

`x^2 = 1 - y^2`  and  `y^2 <=1`

Substituting for `x^2` , we want the area of the bounded region

`f(y) = 65 - 2y^2`

where `0 <= y <=1 `.

Integrating, the bounded region is

`B = int_0^1 (65 - 2y^2) = (65y - 2/3y^3)|_0^1 = 65 -2/3 = 64 1/3`

The area is 64 + 1/3

 

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