INTEGRATION

1. INTEGRATION Gabby Bales, Jake Ellis, and Karey Duane

4. Integration Techniques

6. U-Substitution When you can’t take an anti-derivative easily, use U-Substitution. First, find set u equal to a value in terms of x. Next, Solve for du . Then, plug in u and du into the equation to solve the integral in terms of u. Finally, replace the u - values in the antiderivate with their respective x-values.

7. Example

8. Example

18. Reimannsums

19. Riemann Sums are methods of approximating the area under a curve. This is done by finding the sum of a given number of rectangles or trapezoids drawn under the curve Kinds of Rectangular Sums (identified by point used to find h). • LHS – Left Hand Side • RHS – Right Hand Side • Mid – Midpoint On a graph, the width of the rectangles can be determined by: b-a ---------- (# of rectangles)

20. Example

21. Example 2

22. Example 2

23. Trapezoids

24. Example

25. Fundamental Theory of Calculus

26. The Fundamental Theorem of Calculus (Part 1) Assume that f(x) is continuous on [a, b] and let f(x) be an antiderivative of f(x) on [a, b]. Then:

27. Example 1

28. Example 1

29. The Fundamental Theorem of Calculus (Part 2) Assume that f(x) is a continuous function on [a, b]. Then the area function  is an antiderivative of f(x), that is,…