Homework Homework Assignment #1 Read Section 5.2 Page 308, Exercises: 1 – 25(EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework

Homework Assignment #1 Read Section 5.2 Page 308, Exercises: 1 – 25(EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 3081. An athlete runs with a velocity of 4 mph for a half-hour, 6 mph for the next hour, and 5 mph for another half-hour. Compute the total distance traveled and indicate on a graph how this quantity can be interpreted as an area.


4 6 5

4 0.5 6 1 5 0.5 10.5

d t t t

d mi

0.5 1.0 1.5 2.0 2.5 3.0

Homework, Page 3085. Compute R6, L6, and M3 to estimate the distance traveled over [0, 3] if the velocity at half-second intervals is as follows:


6

6

3

0.5 12 18 25 20 14 20 54.5

0.5 0 12 18 25 20 14 44.5

1.0 12 25 14 51.0

R ft

L ft

M ft

Homework, Page 3089. Estimate R6, L6, and M6 for the function in Figure 15.


6

6

6

0.25 2.35 2.25 2.05 1.6 1.1 0.6 2.4875

0.25 2.4 2.35 2.25 2.05 1.6 1.1 2.9375

0.25 2.38 2.3 2.12 1.85 1.4 0.8 2.7125

R

L

M

Homework, Page 308Calculate the approximation for the given function and interval.


813. , 7 , 3,5R f x x

8

5 3 1 1 28 13 157 3

8 4 4 4 4 41 15 14 13 12 11 10 9 8

4 4 4 4 4 4 4 4 4

92 35

16 4

x

R



2617. , 2 2, 1,4R f x x x

8

4 1 1

6 21.5 2 2.5 3 3.5 4

5 8 12 17 23 30

1 95 15 8 12 17 23 30 47

2 2 2

x

x

f x

R



421. , cos , ,4 2

L f x x

8

2 44 16

4 5 6 7

16 16 16 160.707 0.556 0.383 0.195

0.707 0.556 0.383 0.195 0.36116

x

x

f x

R

Homework, Page 308


4 4

25. Let be the area under the graph of over 0,1 .

Prove that 0.51 0.77 by computing and . Explain

your reasoning.

A f x x

A R L

4 4

4 4

1 0 1

4 41 1 3

0 14 2 4

1 1 30 1

2 22

1 1 1 3 1 1 1 31 0.768 0 0.518

4 2 2 4 2 22 2

Since is an overestimate and is an underestimate, the area

must lie between them.

x

x

f x

R L

R L

x

y


Jon Rogawski

Calculus, ETFirst Edition

Chapter 5: The IntegralSection 5.2: The Definite Integral


Georg Riemann was 19th century German mathematician whose memory is honored by the naming of Riemann Sums.

Riemann Sums are a generalization for the RAM methods discussed in the previous section. The widths may vary and the value of x for which f (x) is found need not be consistent rectangle to rectangle.


The largest subinterval in a Riemann sum is known as the norm ofthe partition.

Example, Page 32116. Describe the partition P and the set of intermediate points C for the Riemann sum shown in Figure 16. Compute the value of the Riemann sum.



If a rectangle falls below the x-axis the sign of its area is negative.

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Homework Homework Assignment #1 Read Section 5.2 Page 308, Exercises: 1 – 25(EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company