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Calculate the surface integral (of function type) where is a surface.
We will solve the (relatively) simple problem: Calculate the surface area of the triangle x+y+z=1, .
Formula Sheet From the formula sheet, we find:
Let the surface S be parameterized by S=(x(u,v),y(u,v),z(u,v)).
Surface element is the magnitude (intensity) of the vector product:
Integrand is the dot product:
Preparation for SOLVER: We must parameterize the surface S with 2 parameters and find the intervals.
Work particular to the given problem:
== Because we want surface area, our function of integration is
== We have an explicit function for the surface . So we just let , and substitue into z: . , i.e. the projection of the triangle onto the u0v = x0y plane.
== So:
To get the intervals: We find the vertices of the triangle in the u0v = x0y plane. In this plane z=0. The vertices of the triangle in 3d are the intersection points of the plane with the postive x, y and z axes (we are given ). So they are (1,0,0), (0,1,0) and (0,0,1). In the u0v plane, we have: V1(u=1,v=0), V2(u=0,v=1) and V3(u=0,v=0). We draw this region. It is a triangle where u goes from 0 to 1 and v goes from 0 to the line joining V1 and V2. The equation of this line is v=-u+1.
SOLVER: Our program follows the hand solution method. Everything in red requires something particular to your problem!
- Parameterize S and input as vector function. Requires parametrization - done above.
- Input f(x,y,z).
- Find the partials of S.
- Find the vector product of the partials. Find the magnitude (intensity) of this vector.
- Substitute parameterization in f.
- Find the integrand.
- Find intervals of integration and integrate. Requires intervals of parametrization - done above.
Step 1: We define , and . Changes with problem; you must input your parametrization in Sf and your function of f.
Step 2: The program finds the partial derivatives.
Step 3: The program calculates
Step 4: The program defines a standard sage function for "change of variables" for a 2-variable parametrization (surface).
The program changes the variables of .
Step 5: Then program finds the dot product of f and dS (the magnitude from step 3).
Step 6: The program computes the integral. Changes with problem - we must put in our intervals of integration.
Answer: The surface area of the triangle with vertices (1,0,0), (0,1,0) and (0,0,1) is: $\frac{\sqrt{3}}{2} \approx 0.87$
The triangle itself is an equilateral triangle with side length so with area
So we expect the surface area to be