Summary - Integration in Two Variables Over a
Rectangular Region
A Riemann sum for f (x, y ) on a rectangle ℛ = [a, b]×[c, d] is a sum of the form
N M
SN,M = ∑ ∑ f Pi j Δxi Δyj
i =1 j =1
corresponding to partitions of [a, b] and [c, d], and choice of sample points Pi j in the subrectangle
ℛi j .
The double integral of f (x, y ) over ℛ is defined as the limit (if it exists):
N M
∫ ∫ℛ f (x, y ) A = lim ∑ ∑ f Pi j Δxi Δyj
M, N→∞ i=1 j=1
We say that f (x, y ) is integrable over ℛ if this limit exists.
Figure 1. The volume of the box is f xi , yj ΔAi j , where ΔAi j = Δxi Δyj .
A continuous function on a rectangle ℛ is integrable.
The double integral is equal to the signed volume of the region between the graph of z = f (x, y ) and
the rectangle ℛ. The signed volume of a region is positive if it lies above the x y-plane and negative
if it lies below the x y-plane.
Rectangular Region
A Riemann sum for f (x, y ) on a rectangle ℛ = [a, b]×[c, d] is a sum of the form
N M
SN,M = ∑ ∑ f Pi j Δxi Δyj
i =1 j =1
corresponding to partitions of [a, b] and [c, d], and choice of sample points Pi j in the subrectangle
ℛi j .
The double integral of f (x, y ) over ℛ is defined as the limit (if it exists):
N M
∫ ∫ℛ f (x, y ) A = lim ∑ ∑ f Pi j Δxi Δyj
M, N→∞ i=1 j=1
We say that f (x, y ) is integrable over ℛ if this limit exists.
Figure 1. The volume of the box is f xi , yj ΔAi j , where ΔAi j = Δxi Δyj .
A continuous function on a rectangle ℛ is integrable.
The double integral is equal to the signed volume of the region between the graph of z = f (x, y ) and
the rectangle ℛ. The signed volume of a region is positive if it lies above the x y-plane and negative
if it lies below the x y-plane.
