1. Trapezoidal Approximation Objective: To relate the Riemann Sum approximation with rectangles to a Riemann Sum with trapezoids.

2. Trapezoidal Approximation We will now look at finding area by using trapezoids rather than rectangles. Remember, the area of a trapezoid is. In the picture below, the height of each trapezoid is. The bases of each trapezoid are the parallel sides. For us, this will be the value of the function evaluate at the endpoints of each “strip.”

3. Trapezoidal Approximation For example, the area of the first trapezoid would be:

4. Trapezoidal Approximation For example, the area of the second trapezoid would be:

5. Trapezoidal Approximation For example, the area of the third trapezoid would be:

6. Trapezoidal Approximation For example, the area of the nth trapezoid would be:

7. Trapezoidal Approximation When adding these areas together and factoring out the common it becomes

8. Trapezoidal Approximation Notice how each value of y is used twice except for the first and last ones. This leads us to another form of the equation.

9. Example Selected values of a continuous function are given in the table. Using 10 subintervals of equal length, the Trapezoidal Rule approximation for is

10. Example Selected values of a continuous function are given in the table. Using 10 subintervals of equal length, the Trapezoidal Rule approximation for is

11. Example Using the subintervals [1, 5], [5, 8], and [8, 10], what is the trapezoidal approximation to ?

12. Example Using the subintervals [1, 5], [5, 8], and [8, 10], what is the trapezoidal approximation to ? Trap 1 Trap 2 Trap 3

13. Example Using the subintervals [1, 5], [5, 8], and [8, 10], what is the trapezoidal approximation to ? Trap 1 Trap 2 Trap 3

14. Example If three equal subdivisions of [0, 3] are used, what is the Trapezoidal Rule approximation of ?

15. Example If three equal subdivisions of [0, 3] are used, what is the Trapezoidal Rule approximation of ? Trap 1 Trap 2 Trap 3