Math 20, Fall 2020 — Grzegrzolka/SchaefferMidterm 1 (January 29th, 2020)

Last/Family Name First/Given Name Seat # Exam #

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Signature: _______________________________________________________Complete problems 1–12. Remember to show your work and justify your answer ifrequired. Present all solutions in as organized a manner as possible. GOOD LUCK!

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Integration table entries you may need. Remember to cite these by roman numeral if you use them.

I.∫

du

u2 + a2=

1

aarctan

(ua

)+ C II.

∫du

u2 − a2=

1

2aln

∣∣∣∣u− a

u+ a

∣∣∣∣+ C

III.∫ √

a2 − u2 du =1

2u√u2 − a2 +

a2

2arcsin

(ua

)+ C IV.

∫du√

a2 − u2= arcsin

(ua

)+ C

V.∫ √

u2 ± a2 du =1

2u√

u2 ± a2 ± a2

2ln∣∣∣u+

√u2 ± a2

∣∣∣+ C

________________________________________________________________________________In problems 1–5 you do not need to justify your answers unless otherwise specified.

1. a. Complete the equation in the statement below:

If f(x) is a differentiable function, then∫ b

a

f ′(x) dx =

b. What is the name of the (completed) statement in (a)?

2. Some function y = f(x) is graphed below.

290 Chapter 5 KEY CONCEPT: THE DEFINITE INTEGRAL

PROBLEMS

23. The graph of ! (") is in Figure 5.40. Which of the fol-lowing four numbers could be an estimate of ∫ 1

0 ! (")#"accurate to two decimal places? Explain your choice.I. −98.35 II. 71.84 III. 100.12 IV. 93.47

0.5 1.0

20

40

60

80

100 ! (")

"

Figure 5.40

24. (a) What is the total area between the graph of ! ($)in Figure 5.41 and the $-axis, between $ = 0 and$ = 5?

(b) What is ∫ 50 ! ($) #$?

3

5

✠

Area = 7

✒

Area = 6

$

! ($)

Figure 5.41

25. Find the total area between % = 4 − $2 and the $-axisfor 0 ≤ $ ≤ 3.

26. (a) Find the total area between ! ($) = $3 − $ and the$-axis for 0 ≤ $ ≤ 3.

(b) Find ∫3

0! ($)#$.

(c) Are the answers to parts (a) and (b) the same? Ex-plain.

In Problems 27–33, find the area of the region between thecurve and the horizontal axis27. Under % = 6$3 − 2 for 5 ≤ $ ≤ 10.28. Under % = cos " for 0 ≤ " ≤ &∕2.29. Under % = ln $ for 1 ≤ $ ≤ 4.30. Under % = 2 cos("∕10) for 1 ≤ " ≤ 2.31. Under % = cos

√$ for 0 ≤ $ ≤ 2.

32. Under the curve % = 7 − $2 and above the $-axis.33. Above the curve % = $4 − 8 and below the $-axis.

34. Use Figure 5.42 to find the values of(a) ∫ '

( ! ($) #$ (b) ∫ )' ! ($) #$

(c) ∫ )( ! ($) #$ (d) ∫ )

( |! ($)| #$

( ' )

! ($)

✠

Area = 13

■Area = 2

$

Figure 5.42

35. Given ∫ 0−2 ! ($) #$ = 4 and Figure 5.43, estimate:

(a) ∫ 20 ! ($) #$ (b) ∫ 2

−2 ! ($) #$(c) The total shaded area.

−2 2−2

2

! ($)

$

Figure 5.43

36. (a) Using Figure 5.44, find ∫ 0−3 ! ($) #$.

(b) If the area of the shaded region is *, estimate∫ 4−3 ! ($) #$.

−4 −3 −2 −1 1 2 3

4

5−1

1

$

! ($)

Figure 5.44

37. Use Figure 5.45 to find the values of(a) ∫ 2

0 ! ($) #$ (b) ∫ 73 ! ($) #$

(c) ∫ 72 ! ($) #$ (d) ∫ 8

5 ! ($) #$

2 4 6 8 10−2−1

12

! ($)

$

Figure 5.45: Graph consists of a semicircle andline segments

Use the graph to compute the values of the definite integrals in the table.

Be extra careful to double-check the endpoints of the integrals.

i.∫ c

a

f(x) dx = ii.∫ c

b

f(x) dx =

iii.∫ b

a

f(x) dx = iv.∫ c

a

|f(x)| dx =

2

3. A hot air balloon moves up and down along a straight line, starting from the ground at t = 0.

The velocity of the balloon is graphed below.

3 5 7 9 11

10

20

30

40

0

–10

–20

1

v (ft/min)

t (min)2 4 6 8 10

The vertical axis (velocity) is measured in feet per minute, with positive velocities corresponding toupward movement and negative velocities corresponding to downward movement. The horizontalaxis (time) is measured in minutes. The graph is of the balloon’s velocity, not its height.

a. At what time is the balloon traveling fastest (in either direction)?

b. At what time is the balloon highest above the ground?

c. How high is the balloon at the time you gave in (a)?

d. At what time(s) is the balloon on the ground? List them all.

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4. The curve y =√25− x2 is graphed below for −5 ≤ x ≤ 5.

-30 -25 -20 -15 -10 -5 0 5

-10

-5

5

a. In the graph above, draw the 5th right Riemann sum for the definite integral∫ 5

0

√25− x2 dx.

b. Will the value of the Riemann sum (that is, the combined area of the rectangles) you drew in(a) be less than, greater than, or equal to the integral’s exact value?Fill in the blank with the correct symbol, <, >, or =.

(value of the R. sum you drew) _____∫ 5

0

√25− x2 dx

c. Evaluate the definite integral∫ 5

0

√25− x2 dx using whatever method you prefer.

You do not need to show your work. Only your final answer (which must be exact, not anapproximation, for full credit) will be graded. Draw a box around your final answer.

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5. Below are four strategies for integration you could potentially use:

I. Integration by substitution with u = x2 II. Integration by substitution with u = x3

III. Integration by parts with u = x IV. Integration by parts with u = x2

In parts (a–e) below, circle one roman numeral I–IV that corresponds to an effective strategy toevaluate that integral; or circle NONE if you believe that none of the strategies above will work.

You do not need to justify your answers and you definitely do not need to evaluate the integrals.

a.∫x cos(x2) dx I II III IV NONE

______________________________________________________________________

b.∫

cos(x2) dx I II III IV NONE______________________________________________________________________

c.∫x2 cos(x3) dx I II III IV NONE

______________________________________________________________________

d.∫x cos(x) dx I II III IV NONE

______________________________________________________________________

e.∫

cos(x3) dx I II III IV NONE

In problems 6–12, evaluate the indefinite integral. Show your work. (See next page for more instructions.)Please draw a box around each of your final answers.

6.∫

3x5 + 4x3 + x+ 5

x2dx

5

In problems 6–12, evaluate the indefinite integral. Show your work. (Problem 6 on previous page.)

• If you use substitution or integration by parts, be sure to clearly specify what you chose for u.• However, if you use a linear substitution like u = Ax+B, you do not need to show every step.• If you use a table entry (these are on page 2), be sure to specify which entry you used, along with

what you chose for u and a.

Please draw a box around each of your final answers.

7.∫

[4 cos(2θ)− 6 sin(3θ)] dθ

8.∫

(7x+ 3)9 dx

6

9.∫

sin(√t)√t

dt

10.∫x4 lnx dx

7

11.∫x2e5x

3+1 dx

12.∫

dx

9x2 + 6x+ 17Hint: (3x+ 1)2 = 9x2 + 6x+ [?].

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DO NOT DETACH THIS (OR ANY OTHER) PAGE.The remaining space is provided for any extra work. If you think this work is important to one of yoursolutions, please indicate that on the page of the relevant problem (otherwise we won’t know to look!).

If you need additional space for work beyond this, ask a proctor.

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DO NOT DETACH THIS (OR ANY OTHER) PAGE.The remaining space is provided for any extra work. If you think this work is important to one of yoursolutions, please indicate that on the page of the relevant problem (otherwise we won’t know to look!).

If you need additional space for work beyond this, ask a proctor.

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