Section 5.5

o Trapezoidal Ruleo Questions on HW

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

21y x 1

8

4

0

y x dx

4

0

y x dx

Solutions: LRAM – 23/4, underestimate

RRAM – 31/4, overestimate

Exact value - 20/3

4

0

y x dx

Trapezoidal Rule

x0=a

y(x)

x1 x2 x3 x4=b

o 1

1 b aT y y

2 n

1 2

1 b ay y

2 n

2 3

1 b ay y

2 n

3 4

1 b ay y

2 n

o 1 2 3 4

1 b aT y 2y 2y 2y y

2 n

o 1 2 3 4

hT y 2y 2y 2y y

2

Trapezoidal Rule:

0 1 2 12 2 ... 22 n n

hT y y y y y

( h = width of subinterval )

2

LRAM RRAMT

Trapezoidal Rule

x0=a x1 x2 x3 x4=b

o 1 2 3 4

hT y 2y 2y 2y y

2

21y x 1

8

Use the trapezoidal

rule with n=4 to approximate

4

0

y x dx

1 9 3 17T 1 2 2 2 3

2 8 2 8

1 27T

2 2

27T

4

Example 1:

Use the trapezoidal rule with n=4 to approximate 4

2

x dx

1 5 7T 2 2 2 3 2 4

4 2 2

1T 24

4

T 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.

t (hours)

R(t) (gallons/hr)

0 9.6

4 10.3

8 10.9

12 11.1

16 10.9

20 10.5

24 9.6

24

0

R t dt

R t 0

2Q t 0.01 950 25t t

Part a

Example 2: #58 p.319

24

0

4R t dt 9.6 2 10.3 2 10.9 2 11.1 2 10.9 2 10.5 9.6

2

253.2 gallons

This value approximates the total number of gallons of water that leaked out of the pipe from t=0 to t=24

Part b

Example 2: #58 p.319

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

R b R aR c

b aR 24 R 0

R c24

R c 0

Part c

Example 2: #58 p.319

24

0

1Average Rate Q t dt

24 0

10.58 gallons per hour