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:
R

m n

ʃ ʃ f(x,y) dA = lim ∑ ∑ f(x_ij*, y_ij*) ∆A
R m,n →∞ i=1 j=1

where (x_ij*, y_ij*) is the sample point within each rectangular area in the xy-plane,
f(x_ij*, y_ij*) is the value of the surface at each sample point, and ∆A is the area of each

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.

The finite sum of each individual "skyscraper" with a rectangular base area, ∆A, and height,

f(x_ij*, y_ij*), yields an approximate value of the volume for each individual "skyscraper".

Click here for an example.

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