Section 5.6: Approximating Sums. Suppose we want to compute the signed area of a region bounded by the graph of a function from x = a to x = b . Let’s try an easier one first. Now, back to the bigger challenge. Maybe we can approximate the area. Maybe we can approximate the area.
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Suppose we want to compute the signed area of a region bounded by the graph of a function from x = a to x = b.
Maybe we can approximate the area...
Let the interval [a, b] be partitioned into n subintervals by any n+1 points
a = x0 < x1 < x2 < … < xn-1 < xn = b
and let xi = xi – xi-1 denote the width of the ith subinterval. Within each subinterval [xi-1, xi], choose any sampling point ci. The sum
Sn = f (c1)x1 + f (c2)x2 + … + f (cn)xn
is a Riemann sum with n subdivisions for f on [a, b].
Let a function f (x) be defined on the interval [a, b]. The integral of f over [a, b], denoted
is the number, if one exists, to which all Riemann sums Sn tend as as n tends to infinity and the widths of all subdivisions tend to zero. In symbols:
For any positive integer n, the quantity
is the Simpson’s rule approximation with 2n subdivisions.