How do you find the surface area of the part of the circular paraboloid z=x^2+y^2 that lies inside the cylinder x^2+y^2=1?

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How do you find the surface area of the part of the circular paraboloid z=x^2+y^2 that lies inside the cylinder x^2+y^2=1?
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  1. Concept of partial derivatives
  2. The area of a surface, ##f(x,y)##, above a region R of the XY-plane is given by ##int int_R sqrt((f_x’)^2 + (f_y’)^2 +1) dx dy## where
    ##f_x’## and ##f_y’## are the partial derivatives of ##f(x,y)## with respect to ##x## and ##y## respectively.
  3. In converting the integral of a function in rectangular coordinates to a function in polar coordinates: ##dx dy rarr (r) dr d theta##

If ##z = f(x,y) = x^2 + y^2##
then ##f_x’ = 2x## and ##f_y’= 2y##

The Surface area over the Region defined by ##x^2+y^2 = 1##is given by
##S =int int_R sqrt(4x^2 + 4y^2 + 1) dx dy##

Converting this to polar coordinates (because it is easier to work with the circular Region using polar coordinates)
##S = int_(theta = 0)^(2pi) int_(r=0)^1 (4 r^2+1)^(1/2) (r) dr d theta##

##= int_(theta=0)^(2pi) ((4r^2+1)^(3/2))/(12) |_(r=0)^1 d theta##

##= int_(theta=0)^(2pi) (5sqrt(5)-1)/(12) d theta##

##= (5sqrt(5) -1)/(12) theta |_(theta=0)^(2pi)##

##= (5sqrt(5)-1)/6 pi##

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