1. Definite Integrals

2. Definite Integral is known as a definite integral. It is evaluated using the following formula Otherwise known as the Fundamental Theorem of Calculus or the Integral Evaluation Theorem

3. Evaluate the following definite integral: Answer: = -6 Answer: = 8.67

4. Evaluate the following definite integral: Answer: = 58 Answer: =

5. Area Under a Curve With this information, we can approximate the area under a curve by drawing rectangles and adding up the area of these rectangles. The more rectangles we use, the better our estimate will be. We can find the area of the region defined by the curve, the x-axis and two vertical lines.

6. Calculate the area under the curve between x = 1 and x = 3.

7. Calculate the area under the curve between x = 1 and x = 5.

8. Riemann Sum is called the Riemann sum of f.

9. Therefore, the area under a curve can be found using integration. Definition of Two Special Definite Integrals

10. To calculate the area under a curve: 1. Draw a quick graph of the curve, or check the curve on a calculator. 2. Use your graph to determine the region which you are evaluating. 3. Regions above the x-axis are positive, regions below the x-axis are negative. 4. Be careful when solving questions that have regions above and below the x-axis as their signs need to be considered.

11. Find the definite integral between the following curve and the x-axis: Answer = -4.5

12. Find the area between the following curve and the x-axis: Answer = 4.5 square units

13. Find the area between the following curve, the x-axis and the line x = 3: Answer = 4 square units

14. Find the area between the following curve and the x-axis from x = 0 and x = 5: Answer =

15. Find the area between the following curve and the x-axis from x = 0 to x = 3: Answer =

16. Discontinuous Functions Although we are able to evaluate this does not mean that it will have any geometrical importance. A function must be defined and continuous in order for the definite integral to have meaning. Example: has no meaning since it is not defined at x = 0.

17. Properties of Definite Integrals These two properties are useful when calculating the area between two curves.

18. Find the area between the following curves: Answer = and

19. Find the area between the following curves: Answer = and

20. Find the area between the following curves from x = 0 and x = 1. Answer = and

21. Find the area between the following curves: Answer = and

22. Find the shaded area between the following curves: Answer = and

23. Find the area enclosed by the curve below, the y-axis, and the lines y = 1 and y = 8: Answer =

24. Find the area enclosed by the curve below, the y-axis, and the lines y = -2 and y = 3: Answer =

25. Find the area between the following curve and the x-axis: Answer =

26. Find the total area between the following curve and the x-axis from x = 1 to x = 3: Answer = The region enclosed by the curves and, where k > 0, is denoted by R. Given that the area of R is 12, find the value of k. Answer: k = 6

27. Find the area enclosed between the following curves: Answer = and

28. Find the area enclosed between the following curves: Answer = and

29. Find the area between the following curves: Answer = and