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.
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.