1. 1 5.2 – The Definite Integral

2. 2 Review Evaluate

3. 3 The Definite Integral 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 ].

4. 4 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.

5. 5 Left, Right, Midpoint Rules Left: Right: Midpoint:

6. 6 Examples 1. Use the Midpoint Rule 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.

7. 7 Examples 3. Use the form of the definition of the definite integral to evaluate the integral. 4. Express the integral as a limit of a Riemann Sum, but do not evaluate. 5. Evaluate the integrals by interpreting in terms of area.

8. 8 Basic Properties of the Integral Let a, b, and c be constants and f and g be continuous functions on [a, b]. 3. Definite integrals can be positive or negative. 4. Not all definite integrals can be interpreted in terms of area, but definite integrals can be used to determine area.

9. 9 Properties of the Integral Let a, b, and c be constants and f and g be continuous functions on [a, b].

10. 10 Properties of the Integral Let a, b, and c be constants and f and g be continuous functions on [a, b]. 5. If f (x) ≥ 0 for a ≤ x ≤ b, then 6. If f (x) ≥ g (x) for a ≤ x ≤ b, then 7. If m ≤ f (x) ≤ M for a ≤ x ≤ b, then

11. 11 Examples 6. Use the properties of integrals to verify the inequality without evaluating the integrals.