Question

Find the surface area of the portion of the Parabloid z=x² + y² below the plane z=1​

Answer 1

Answer:

Concept of partial derivatives

The area of a surface, f(x,y), above a region R of the XY-plane is given by ∫∫R√(fx')2+(fy')2+1dxdy where

fx' and fy' are the partial derivatives of f(x,y) with respect to x and y respectively.

In converting the integral of a function in rectangular coordinates to a function in polar coordinates: dxdy→(r)drdθ

If z=f(x,y)=x2+y2

then fx'=2x and fy'=2y

The Surface area over the Region defined by x2+y2=1is given by

S=∫∫R√4x2+4y2+1dxdy

Converting this to polar coordinates (because it is easier to work with the circular Region using polar coordinates)

S=∫2πθ=0∫1r=0(4r2+1)12