Week #10 - The Integral Section 5

[Pages:6]Week #10 - The Integral Section 5.2

From "Calculus, Single Variable" by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc.

This material is used by permission of John Wiley & Sons, Inc.

SUGGESTED PROBLEMS 1. Using Figure 5.24, draw rectangles representing each of the following Riemann sums for the function f on the interval 0 t 8. Calculate the value of each sum.

(a) Left-hand sum with t = 4 (b) Right-hand sum with t = 4 (c) Left-hand sum with t = 2 (d) Right-hand sum with t = 2

Figure 5.24 (a) Left-hand sum =32 ? 4 + 24 ? 4 = 224.

0 4 8 16 24 32

0

2

4

6

8

t

(b) Right-hand sum = 24 ? 4 + 0 ? 4 = 96.

0 4 8 16 24 32

0

2

4

6

8

t

(c) Left-hand sum = 32 ? 2 + 30 ? 2 + 24 ? 2 + 14 ? 2 = 200.

1

0 4 8 16 24 32

0

2

4

6

8

t

(d) Right-hand sum = 30 ? 2 + 24 ? 2 + 14 ? 2 + 0 ? 2 = 136.

0 4 8 16 24 32

0

2

4

6

8

t

15

3. Use Figure 5.26 to estimate f (x)dx.

-10

Figure 5.26 We know that

15

f (x)dx = Area under f (x) between x = -10 and x = 15

-10

The area under the curve consists of approximately 14 boxes, and each box has area (5)(5) = 25. Thus, the area under the curve is about (14)(25) = 350, so

15

f (x)dx 350

-10

In Exercises 9-11, use a calculator or a computer to find the value of the definite integral to four decimal places.

3

9. 2xdx

0

The extra instruction in the assignment asked for 6 intervals to be used, with the left-hand sum. If 6 intervals are used, over the range x = 0 to x = 3, then x = 3/6 = 0.5.

2

x 2x

0.0 0.5

2(0.0) 2(0.5)

= 1 =2

1.0 1.5

2(1.0) 2(1.5)

= 2 =2 2

2.0 2.5

2(2.0) 2(2.5)

= 4 =4 2

The left-hand sum would be

3

f (x)dx x(f (0) + f (0.5) + . . . + f (2.5))

0

= (0.5)(1 + 2 + 2 + 2 2 + 4 + 4 2)

=

7

+7 2

2 8.45

In Exercises 12-18, find the area of the regions between the curve and the horizontal axis

16. Under y = cos(t/10) for 1 t 2.

This question was included in the original courseware package by mistake: there is no part (c) to answer.

17. Under the curve y = 7 - x2 and above the x-axis.

The extra instruction in the assignment asked for 6 intervals to be used, with the left-hand sum. To begin with, we need to find where the graph of 7 - x2 intersects the x-axis, or when y = 0:

7 - x2 = 0

x=? 7

7

So we want to estimate

7 - x2dx, using six intervals and the left-hand sum. To do

-7

this, we'll need the following values of the function:

x

-

-

2 3

-

1 3

7 7

7

1

3 2 3

0 7

7

7 - x2 0

3.8889 6.2222

7 6.2222 3.8889

Compute the left-hand sum by

3

7 3

?

(0

+

3.8889

+

6.2222

+

7

+

6.2222

+

3.8889)

=

24.00774

QUIZ PREPARATION PROBLEMS

3

2. Use Figure 5.25 to estimate f (x)dx.

0

Figure 5.25

We estimate the area by counting the rectangles below the graph. There are 3 full and about 4 partial rectangles, for a total of approximately 5 rectangles. Since each rectangle represents 4 square units, our estimated area is 5(4) = 20. We have

3

f (x)dx 20

0

19. (a) On a sketch of y = ln(x), represent the left Riemann sum with n = 2 approximating

2

ln xdx. Write out the terms in the sum, but do not evaluate it.

1

(b) On another sketch, represent the right Riemann sum with n = 2 approximating

2

ln xdx. Write out the terms in the sum, but do not evaluate it.

1

(c) Which sum is an overestimate? Which sum is an underestimate?

(a) You can't see the first rectangle because it has zero height.

1

1.5

2

x

The left-hand sum is therefore (0.5) ? (ln(1) + ln(1.5)) = ln(1.5) ? 0.5 0.2027 (b) The right sum is easier to see.

4

1

1.5

2

x

The right-hand sum is therefore (0.5) ? (ln(1.5) + ln(2)) = 0.5493

(c) The right-hand sum is an overestimate, while the left-hand sum is an underestimate.

27. Use Figure 5.29 to find the values of

b

(a) f (x)dx

a c

(b) f (x)dx

b c

(c) f (x)dx

a c

(d) |f (x)|dx

a

Figure 5.29

(a) The area between the graph of f and the x-axis between x = a and x = b is 13, so

b

f (x)dx = 13.

a

(b) Since the graph of f (x) is below the x-axis for b < x < c,

c

f (x)dx = -2.

b

(c) Since the graph of f (x) is above the x-axis for a < x < b and below the x-axis for b < x < c,

c

f (x)dx = 13 - 2 = 11.

a

5

(d) The graph of |f (x)| is the same as the graph of f (x) except that the part below the x-axis is reflected to be above it. Thus

c

|f (x)|dx = 13 + 2 = 15.

a

31. Using the graph of 2 + cos x, for 0 x 4, list the following quantities in increasing

4

order: the value of the integral (2 + cos x)dx, the left sum with n = 2 subdivisions,

0

and the right sum with n = 2 subdivisions.

The two rectangles for both the left- and right-hand sums are both of height 3, as shown in the diagram below.

3

0

0

2

4

x

Thus, the actual area beneath the curve is smaller than both the left- and right-hand sums. The left- and right-hand sums are equal to each other.

32. Sketch the graph of a function f (you do not need to give a formula for f ) on an interval [a, b] with the property that with n = 2 subdivisions,

b

f (x)dx < Left-hand sum < Right-hand sum

a

The easiest way to answer this question is to try drawing graphs and the corresponding left- and right-hand sums. After a while, you notice that for the actual area to be below the left- and right-hand sums, the function must dip down between the end points, and the left end points must be lower than the right end points.

A sample graph is shown below. The left graph shows the rectangles for the left-hand sum, while the right graph shows the (larger area) from the right-hand sum. The actual area under the graph is smaller than either rectangular approximation.

0 x

0 x

6

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download