# Math 131 Fall 2021 Exam 1

Math 131 Fall 2021 Exam 1

Name

Make sure to show all work to receive full credit for each question. Answers without supporting work will receive a score of zero. Also, please box your final answers, do your best to write clearly and legibly, and write your name on every odd-numbered page.

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1. (1 pt each) Below are 4 graphs, each illustrating a Riemann Sum.

(a) (b)

(c) (d)

Match the graph above with its Riemann Sum below.

5∑ i=1

f(xi−1) ∆x 5∑ i=1

f(xi) ∆x 5∑ i=1

f(x̄i) ∆x 5∑ i=1

f(x∗i ) ∆x

2. (6 pts) Circle the letter next to each statement that is always true.

(a) ∫ cf(x)dx = c

∫ f(x)dx where c is a constant

(b) ∫

[f(x)g(x)] dx = (∫

f(x)dx )(∫

g(x)dx )

(c) ∫ c a f(x) dx +

∫ b c f(x) dx =

∫ b a f(x) dx

(d) ∫

f(x) g(x)

dx = ∫ f(x)dx∫ g(x)dx

(e) ∫

[f(x) ±g(x)] dx = ∫ f(x)dx±

∫ g(x)dx

(f) ∫ b a f(x)dx = −

∫ a b f(x)dx

/ 10 pts

1

3. (1 pt) The function f(x) = sin(x) is continuous on what interval?

(a) [−π,π] (b) [0,π]

(c) [−π, 0] (d) (−∞,∞)

4. (1 pt) Circle the phrase that best completes the statement below:

The equation

∫ x=1 x=−1

ln(x) dx = [x ln(x) −x]x=1x=−1 is

(a) correct by the Fundamental Theorem of Calculus Part II.

(b) incorrect because the antiderivative of f(x) = ln(x) is F(x) = 1 x

, not x ln(x) −x. (c) incorrect because 1

x is discontinuous at x = 0.

(d) incorrect because ln(x) is discontinuous on [−1, 0].

5. (2 pt each) Match the expression on the left with its definition on the right.

x̄i

∑n i=1 f(xi) ∆x

∆x

xi−1

lim n→∞

∑n i=1 f(xi) ∆x

xi

x∗i

(a) a + i∆x

(b) ∫ b a f(x) dx

(c) ith subinterval midpoint

(d) Net area sum of n approximating rectangles

(e) b−a n

(f) ith subinterval sample point

(g) ith subinterval left endpoint

6. (2 pts) Consider points C and D. One is the centroid of the shaded region; the other is the average midpoint of n approximating rectangles as n → ∞. Label each point, and briefly explain your choices in the section below right.

/ 18 pts

2

Integration

7. (10 pts each) Solve both integrals:

(a) ∫ [

1 x3

+ cos(2x) + 1 x

] dx (b)

∫ e √ x dx

/ 20 pts

3

8. (10 pts) Choose either (a) or (b) below to solve. Do not solve both.

(a) ∫ 2π 0

sin( 1 3 x)2 dx

/ 10 pts

4

(b) ∫

2×2+2 2×2+5x+3

dx

/ 10 pts

5

Applications

9. (10 pts) Chose either (a) or (b) below to solve. Do not solve both.

(a) A particle moves along a line with velocity function v(t) = t2−t, where v is measured in meters per second. Find the displacement and the distance traveled by the particle during the time interval [0, 5]

/ 10 pts

6

(b) Find the centroid of the region bound by and f(x) = −x2 + 4 and the x-axis

To receive full credit: 1) Graph the region below; 2) Locate and graph the points of intersection of f and the x-axis; 3) Find the coordinates of the centroid and graph it below.

/ 10 pts

7

Work

A bucket that weighs 4 lbs and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lbs of water and is pulled up at a rate of 2 ft/sec, but water leaks out of hole in the bucket at a rate of .2 lbs / sec. Find the work done in pulling the bucket to the top of the well.

10. (1 pt) If F measures force and d distance, circle the equation below that measures Work (W ):

(a) W = F d

(b) W = Fd

(c) W = F + d

(d) W = F −d

11. (1 pt) The units of force acting on the water and bucket as it rises to the top of the well are

(a) Newtons (N)

(b) Joules (J)

(c) Pounds (lbs)

(d) Foot-Pounds (ft. – lbs)

12. (1 pt) The units of work required to lift the bucket and water to the top of the well are

(a) Newtons (N)

(b) Joules (J)

(c) Pounds (lbs)

(d) Foot-Pounds (ft. – lbs)

13. (1 pt) Choose the answer that best fills in the blanks.

As the bucket and water move up the well, the force acting on bucket is because its weight is , but the force acting on the water is because its weight is .

(a) constant, constant, changing, changing

(b) constant, changing, changing, constant

(c) changing, constant, changing, constant

(d) changing, changing, constant, constant.

14. (3 pts) Based on your answer to 13 above, calculate the work W needed to haul the bucket alone from the bottom to the top of the well.

/ 7 pts

8

Now, let’s figure out the work required to haul the water all the way up the well.

15. (1 pt) If t is measured in seconds, which expression is equal to the weight in lbs. of the water in the bucket at time t as it moves up the well?

(a) 40 .2t

(b) 40(.2t)

(c) 40 + .2t

(d) 40 − .2t

16. (1 pt) If x is a measure of feet, which expression gives the distance of the water in the bucket from the bottom of the well at time t?

(a) x = 2 t

(b) x = 2t

(c) x = 2 + t

(d) x = 2 − t

17. (2 pts) Use your answers to questions 15 and 16 above to define an expression for the weight of the water in terms of x.

Scratch work to help you with Questions 15 and 16. Not necessary to use, not counted for credit!

/ 4 pts

9

To find the work W required to haul the water 80 ft up the well, we start by subdividing the 80 ft interval into n equally spaced subintervals of ∆x feet.

When ∆x is very small, the weight of the water is relatively constant throughout the length of the interval.

18. (2 pts) Define an expression that approximates Wi, the work required to lift the water a distance of ∆x feet, from xi−1 to xi.

By summing up all the W ′is, we can approximate the work needed to haul to water from the top to the bottom of the well.

19. (2 pts) Write an expression for the sum of all the W ′is.

We can make this approximation precise by letting n →∞ and then taking the limit.

20. (4 pts) Write an expression that calculates the precise amount of work needed to haul the water from the bottom to the top of the well, and then solve it.

21. (1 pts) Use your answers to Questions 14 and 20 to calculate the total amount of work needed to haul the water and bucket from the bottom to the top of the well.

/ 8 pts

10

Extra Credit: (3 pts) Which of the following areas are equal? To receive credit, your answer must be justified by the correct explanation.

(e) (f)

/ 3 pts

11

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