1 5.e – The Definite Integral as a Limit of a Riemann Sum (Numerical Techniques for Evaluating Definite Integrals)

2 Definitions: Sequence and Series A sequence a n (called the general term) is a function whose domain is positive integers. A series (sigma notation) is the sum of the terms of a sequence.

3 Definition: Series Formulas Let c be a constant and n a positive integer.

4 Series Formulas 2. Write a formula for the series in terms of n: 3. Show that 1. Evaluate:

Area Estimation Using Rectangles |a|a |b|b A1A1 A2A2 A3A3 A4A4 x1x1 x2x2 x3x3 x4x4 Generally x i can be selected randomly from each subinterval, but it is easer to use the left-hand side, right-hand side, or midpoint of each subinterval.

6 Area Estimation Using Rectangles Most generally, x i can be any point within each subinterval. It is impossible to find patterns to random numbers, so x i is always chosen to be on the left-hand side, right-hand side, or midpoint of each subinterval. If given the choice, the right-hand side is always easiest.

7 Area Estimation Estimate the area bounded by the curve y = 25 – x 2, the lines x = 0 and x = 5, and the x-axis? Use the website estimate the area using n = 6 rectangles taken at the right endpoint of each subinterval. Repeat this for n = 12, 24, 48, and 96 rectangles. (a) Which x i gives the best estimate for a small number of rectangles? (b) How can we improve our estimate of the area? Conclusion:

8 The Definite Integral as The Limit of a Riemann Sum If f is a continuous function defined for a ≤ x ≤ b. |a|a |b|b ||| ●x1*●x1* ●x2*●x2* ●x3*●x3* ●x4*●x4* Divide the interval [a, b] into n subintervals of equal width of Δx = (b – a) / n. Let x 1 *, x 2 *, …., x n * be random sample points from these subintervals so that x i * lies anywhere in the subinterval [x i-1, x i ]. The * designates random sample points.

9 The Definite Integral |a|a |b|b ||| ●x1*●x1* ●x2*●x2* ●x3*●x3* ●x4*●x4* Then the definite integral of f from a to b is … Riemann Sum [Bernhard Riemann (1826 – 1866)] The limit of a Riemann Sum as n → ∞ from x = a to x = b. Note: x i is usually taken on the left side, right side, or midpoint of each rectangle since it is impossible to find a pattern for random points. The limit of a Riemann Sum as n → ∞ from x = a to x = b.

10 Examples 1. Use the midpoint of each sub interval with the given value of n to approximate the integral. Round your answer to four decimal places. 2.Write the limit of a Riemann Sum as a definite integral on the given interval.

11 Examples 4. Use the limit of a Riemann Sum definition of the definite integral to evaluate the definite integral. 3. Express the integral as a limit of a Riemann Sum using the left-hand side of each subinterval, but do not evaluate.