Riemann sums:
The value of a definite integral (the area under a curve) can be determined by
calculating the sum of rectangles fit under the curve with progressively smaller
widths.
A left Riemann sum is when the left corner of the rectangles are fit to the curve. The
opposite is true for a right Riemann sum, and the middle is fit for a midpoint Riemann
sum.
For a curve f ( x ) , the integral is defined as:
b n
∫ f ( x ) dx= n→
lim ∑ f ( x i )∆ x where the rectangles used are infinitely narrow.
∞
a i=1
Double integration:
The volume bounded by a function of 2 variables f ( x , y ) can be determined by taking
a Riemann sum using rectangular prism.
The volume of the prism is given by:
V =f ( x i , y j ) ∆ x ∆ y
For a function f ( x , y ), the integral bound by R=[ a , b ] ×[c , d] is divided into infinitely
many rectangular prisms, then:
❑ n m
∬ f ( x , y ) dA= lim ∑ ∑ f (xi , y j )∆ x ∆ y
n , m →∞ i=1 j=1
R
Iterated integrals:
For the above integral definition:
(∫ )
❑ b d b d
∬ f ( x , y ) dA=∫∫ f ( x , y ) dy dx =∫ f ( x , y ) dy dx
R a c a c
Double integrals can be permutated according to Fubuni’s theorem, i.e.:
b d d b
∫∫ f (x , y )dy dx=∫∫ f ( x , y)d x d y
a c c a
Fubuni’s theorem can make double integrals much easier to solve.
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