A slice is made parallel to the base of a right rectangular pyramid, as shown.
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The value of the given surface integral is 4.
The given plane intercepts the coordinate axes at (2, 0, 0), (0, 2, 0), and (4, 0, 0). These point are the coordinates of a triangular region that we can parameterize using.
A surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. Given a surface, one may integrate a scalar field over the surface or a vector field.
with 0≤u≤1 and 0≤v≤1. Then the surface element ds is equivalent to
The surface integral is then
Therefore the value of the given surface integral is 4.
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