1. In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition of the definite integral.

2. Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].

3. The area of this region is not able to be found using basic geometric formulas like we did in section 5.1

4. Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3]. The area of this region is not able to be found using basic geometric formulas like we did in section 5.1 Instead, we estimate the value of the area by using some number of rectangles (left, right, or midpoint) or trapezoids.

5. 1. Divide the interval [a, b] into n subintervals each of width 2. For L n or R n, use the appropriate point from each subinterval (called x i ). The height of each rectangle is and its area is. 3. For M n, use the midpoint of each subinterval (called m i ). The height of each rectangle is and its area is. 4. Add together the area of the n rectangles to get the desired approximation.

6. 1. Divide the interval [a, b] into n subintervals each of width 2. In the i th subinterval, use both of the endpoints of that subinterval, x i and x i+1. The area of this trapezoid is. 1. Add together the area of the n trapezoids to get the desired approximation.

7. Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3]. We use the endpoints:

8. Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3]. We use the endpoints:

9. Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3]. We use the midpoints:

10. Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3]. We use all the endpoints:

11. Find the L 4, R 4, M 4, and T 4 approximation for

12. What can we do to get more accurate approximations?

13. What can we do to get more accurate approximations? Use midpoint rectangles. They are, in general, more accurate than left or right rectangles or trapezoids.

14. What can we do to get more accurate approximations? Use midpoint rectangles. They are, in general, more accurate than left or right rectangles or trapezoids. Use a larger value of n The more shapes used, the more accurate the approximation

15. The definite integral of f from x = a to x = b is formally defined by: