1. LECTURE 8
The Definite Integral

2. Objectives Riemann Sums Definite Integral The Fundamental Theorem
Improper Integrals

3. Area Under a Graph
If f(x) is a continuous non-negative function on the interval a ≤ x ≤ b, we refer to the area bounded by the graph of f(x), the horizontal axis and two vertical lines x = a and x = b as the “area under the graph of f(x) from a to b”.

4. Area Under a Graph

5. Areas and Riemann Sums
Estimate the area: We can, to any accuracy, by constructing rectangles whose total area is approximately the same as the area to be computed.
When the rectangles are thin, the mismatch between the rectangles and the region under the graph is quite small.
Rectangle approximation can be made as close as desired to the exact area simply by making the width of the rectangles sufficiently small.

6. Areas and Riemann Sums

7. Areas and Riemann Sums

8. Riemann Sums Divide the interval [a, b] into n equal subintervals
The width of each subinterval is ∆x=(b-a)/n
In each interval, select a point xi.
Construct ith rectangle with height = f(xi).
Area= f(x1) ∆x + f(x2) ∆x + … + f(xn) ∆x
The sum is called a Riemann sum.

9. Example 1
Use a Riemann sum with n = 4 to estimate the area under the graph of f(x) = x2 from 1 to 3. Select the left end points of the subintervals.

10. Example 2
To estimate the area of a 100-meter-wide waterfront lot, a surveyor measured the distance from the street to the waterline at 20m intervals, starting 10 m from one corner of the lot. Use the data to construct a Riemann sum approximation to the area of the lot (39, 46, 44, 40, 45).

11. Example 2

12. Example 3 The velocity of a rocket at time t is v(t) m/s.
Construct a Riemann sum that estimates how far the rocket travels in the first 10 seconds.
What happens when the number of subintervals in the partition increases without bound?

13. Example 3

14. Definite Integrals f(x) is a continuous nonnegative function on [a,b]
The limit of the Riemann sum is called the “definite integral of f(x) from a to b”
The definite integral of a nonnegative function f(x) equals the area under the graph of f(x).

15. Definite Integrals
𝑎 𝑏 𝑓 𝑥 𝑑𝑥 =𝐵+𝐷−(𝐴+𝐶)

16. Fundamental Theorem of Calculus
Suppose that f(x) is continuous on the interval
a ≤ x ≤ b, and let F(x) be an anti-derivative of f(x).
Then

17. Properties of the Definite Integral

18. Properties of the Definite Integral

19. Integration by Substitution
− 𝑥 2 𝑑𝑥

20. Integration by Substitution
0 1 𝑑𝑥 𝑒 𝑥 +1

21. Integration by Parts
1 2 (𝑥+2) 𝑒 𝑥 𝑑𝑥

22. Integration by Parts
1 2 𝑥+2 𝑙𝑛𝑥𝑑𝑥

23. Example
Some food is placed in a freezer. After t hours the temperature of the food is dropping at the rate of r(t) degrees per hour. Where r(t) = /(t+3)2
Compute the area under the graph of y = r(t) over the interval 0 ≤ t ≤ 2.
What does the area represent?

24. Example
Suppose p(t) is the rate at which pollutants are discharged into a lake, where t is the number of years since Interpret the meaning of the definite integral from 5 to 7 of p(t).

25. Improper Integrals
Either or both of the limits of integration has an infinitive value, we call this an improper integral of the first kind.
The function f(x) is unbounded at one or more of the points in [a,b]. These point are called singularities of f(x). This is called an improper integral of the second kind.
If both conditions are satisfied we now have an improper integral of the third kind.

26. Example
𝟏 ∞ 𝟏 𝒙 𝟐 dx