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April 21st, 2017, 06:19 AM  #1 
Newbie Joined: Nov 2016 From: Louisville, Ky Posts: 7 Thanks: 0  Unoriented Surface Integral Calculus III
Hello, I need help with this problem!  Use the unoriented surface integral to express the mass of a dome in the shape of the paraboloid z= 25x^2y^2 , that lies above the xyplane with density function f(x,y,z)=x^2+y^2 as a double integral over a suitable region D in the xyplane. Evaluate the integral. Thank you! 
April 21st, 2017, 01:59 PM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,340 Thanks: 588 
The mass of a twodimensional object, S, given a density function f(x,y), is just the integral of that density function over S: where "dS" is the "differential of surface area" in terms of x and y. One way to do that is to find dS is first to find the vector differential, . Then then scalar differential of surface area is the magnitude of that vector: . If a surface is given in parametric equations, x= f(u, v), y= g(u, v), z= h(u, v) then the two vectors and lie in the tangent plane and their cross product, is perpendicular to the surface and is so its length is dS. Here, you are given the surface as z= f(x, y) so you can take x and y as the parameters u and v: x= u, y= v, z= f(u, v). Then the vector derivative with respect to u is , the vector derivative with respect to v is and their cross product is and . Last edited by greg1313; April 21st, 2017 at 02:13 PM. 
April 23rd, 2017, 07:03 AM  #3  
Newbie Joined: Nov 2016 From: Louisville, Ky Posts: 7 Thanks: 0  Quote:
 
April 23rd, 2017, 08:03 AM  #4  
Newbie Joined: Nov 2016 From: Louisville, Ky Posts: 7 Thanks: 0  Quote:
 

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calculus, iii, integral, surface, unoriented 
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