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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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section 5 6 approximating sums
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.

slide3

Now, back to the bigger challenge...

Maybe we can approximate the area...

definition of a riemann sum
Definition of a Riemann Sum

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].

commonly used riemann sums
Commonly Used Riemann Sums
  • Left-hand
  • Right-hand
  • Midpoint
the definite integral as a limit
The Definite Integral as a Limit

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:

trapping the integral part i
Trapping the Integral, Part I
  • Suppose f is monotone on [a, b]. Then, for any positive integer n,
  • If f is increasing,
  • If f is decreasing,
trapping the integral part ii
Trapping the Integral, Part II
  • For any positive integer n,
  • If f is concave up on [a, b],
  • If f is concave down on [a, b],
simpson s rule
Simpson’s Rule

For any positive integer n, the quantity

is the Simpson’s rule approximation with 2n subdivisions.