Section 5.5

1. Section 5.5 Trapezoidal Rule Questions on HW

2. Warm-Up • Let • Approximate using LRAM w/ n=4 • Is LRAM an overestimate or underestimate? • Approximate using RRAM w/ n=4 • Is RRAM an overestimate or underestimate? • Evaluate • Solutions:LRAM – 23/4, underestimate • RRAM – 31/4, overestimate • Exact value - 20/3

3. Trapezoidal Rule: ( h = width of subinterval ) Trapezoidal Rule y(x) x0=a x1 x2 x3 x4=b

4. Trapezoidal Rule • Use the trapezoidal rule with n=4 to approximate x0=a x1 x2 x3 x4=b

5. Example 1: • Use the trapezoidal rule with n=4 to approximate

6. Example 2: #58 p.319 • The rate at which water flows out of a pipe is given by a differentiable function R of time t. The table below records the rate at 4-hour intervals for a 24-hour period. • a)Use the Trapezoidal Rule with 6 subdivisions of equal length to approximate . Explain the meaning of your answer in terms of water flow, using correct units. • b)Is there some time t between 0 and 24 such that . Justify your answer. • c)Suppose the rate of water flow is approximated by Use Q(t) to approximate the average rate of water flow during the 24 hour period.

7. Example 2: #58 p.319 • Part a • This value approximates the total number of gallons of water that leaked out of the pipe from t=0 to t=24

8. Example 2: #58 p.319 • Part b • Yes. Since R(t) is differentiable, it is also continuous. By the Mean Value Theorem, there is a number c between 0 and 24 such that

9. Example 2: #58 p.319 • Part c