Approximate
Volumes under a Surface using the Riemann Sum
In a Nut Shell: Premise The volume of a solid S that lies underneath
the surface, f(x,y), and above the rectangle, R, in
the xy-plane can be approximated by the Riemann
Double Sum. Volume = ʃ
ʃ f(x,y) dA is replaced by the double sum as follows:
m n ʃ
ʃ f(x,y) dA = lim ∑ ∑ f(xij*,
yij*) ∆A where (xij*,
yij*) is the sample point within each
rectangular area in the xy-plane, rectangle in the xy- plane. The
location of the sample points will impact the value of the volume determined. One possible sample point is the mid-point
of each rectangular area in the xy-plane. But
other sample points may be taken as well.
See the figure below. |
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The finite sum of each
individual "skyscraper" with a rectangular base area, ∆A, and
height, f(xij*,
yij*), yields an approximate value of
the volume for each individual "skyscraper". |
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Copyright © 2018 Richard C. Coddington
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