Math 132, Exam 1, Fall 2017 Solutions
You may have a single 3x5 notecard (two-sided) with any notes you like. Otherwise you should have only your pencils/pens and eraser. No calculators are allowed. No consultation with any other person or source of information (physical or electronic) is allowed during the exam. All electronic devices should be turned off.
Part I of the exam contains questions 1-14 (multiple choice questions, 5 points each) and true/false questions 15-19 (1 point each). Your answers in Part I should be recorded on the “scantron” card that is provided. If you need to make a change on your answer card, be sure your original answer is completely erased. If your card is damaged, ask one of the proctors for a replacement card. Only the answers marked on the card will be scored.
Part II has 3 “written response” questions worth a total of 25 points. Your answers should be written on the pages of the exam booklet. Please write clearly so the grader can read them. When a calculation or explanation is required, your work should include enough detail to make clear how you arrived at your answer. A correct numeric answer with sufficient supporting work may not receive full credit. Part I
1. Suppose that If and , what is ?
A) B) C) D) E) F) G) H) I) J)
so
So and therefore
2. What are all the value(s) of that make the following equation true?
A) B) C) D) E) F) G) H) I) J) all values of make the equation true
For any ,
3. If , then what is ? A) B) C) D) E) F) G) H I) J)
so 4. Find the area between the graphs of the functions sin and cos .
A) B) C) D) E) F) G) H) I) J)
cos sin sin cos
5. Water is flowing out of the bottom of a tank at a rate of Liters/minute (for . How much water flows out during the first minutes ?
A) L B) L C) L D) L E) L
F) L G) L H) L I) 1300 L J) L
Let be the amount of water in the tank at time Then The change in during the first seconds ln 6. Find the value of A) B) C) D) E)
F) G) H) I) J)
Substitute ln When when , ln So 7. The sum of integrals gives the area of a certain region which can also be calculated as What is this integral?
A) B) C) D) E) F) G) H) I) J)
is the area of the green (lower) region. is the area of the gray (upper) region. Write the upper and lower boundary curves as functions of
and
Then the area of the whole region is given by one integral:
8. If what is ? A) B) C) D) E)
F) G) H) I) J)
Substitute (and When when So
Therefore
9. The shaded region below is revolved around the -axis to create a solid. Using the “disk method ” which integral gives the volume of the solid?
A) B) C) D) E) F) G) H) I) J) The upper boundary is ; the lower boundary is the constant function The integral is outer radius of washer inner radius of washer 16 10. Suppose is differentiable and that 8 Find the value of A) B) C) D) E) F) G) H) I) J)
Substitute . When when . Then
11. The shaded region pictured below is revolved around the -axis to create a solid. Using the “shell method ” which integral gives the volume of the solid?
A) B) C) D) E) F) G) H) I) J) The upper boundary of the region is and the lower boundary is For revolution around the -axis, using the shell method, is 12. What is ? A) B) C) D) E) F) G) H) I) J)
Substitute When when
Then So .
13 It is a fact that Use this fact and the substitution to evaluate A) 0) B) C) D) E)
F) G) H) I) J)
Substitute (so When when
So 4 The base of a solid is the circular disk whose boundary is a circle with radius inches, centered at the origin. All the cross sections of the solid perpendicular to the - axis are squares. (See drawing.)
Which of the following integrals (after a bit of simplification) gives the volume of the solid?
A) B) C) D) E) F) G) H) I) J)
The boundary of the base consists of two functions: above the axis and below the axis). The cross section at is a square whose sides have length so the cross sectional area at is Therefore the volume is Questions 15-19 are true/false:
15. The function has an antiderivative.
A) True B) False
True: every continuous function has an antiderivative For example, in this case (by Fundamental Theorem of Calculus, Part I), is an antiderivative for because
16. cos 0 cos A) True B) False
True: if cos then cos cos so is an even function (graph is symmetric with respect to the -axis). For an even function So cos cos 0 cos 17. Let be the triangular region bounded by the -axis, the -axis, and the line . When is revolved around the -axis, you will get the same volume as when is revolved around the -axis.
A) True B) False
False: Revolving around the -axis gives a cone with Revolving around the -axis gives a cone with These cones have different volumes: ) You should know the formula for the volume of a cone; but if you didn't, you could write the two simple integrals for the volumes and compare those.
18. The substitution changes the integral into A) True B) False
If then =
19. A) True B) False
If then so is an odd function (graph is symmetric with respect to the origin). So PART II WRITE YOUR ANSWERS IN THE SPACE PROVIDED
20. You want to find the area between the graphs of and a) On the grid below, draw the graphs and find the values where the curves intersect . Mark the intersection points on the graphs. In the box below, write the integral that gives the area.
The curves intersect where vv Area Integral: , this simplification not required) b) Evaluate the integral. Put your final answer in the box.
Area Value of integral 21. The region bounded by and is revolved around the -axis
a) Write an integral that gives the volume generated when the region is revolved around the -axis. Put your answer in the box. You do not need to evaluate the integral.
The two graphs intersect where The intersection point in the region is at
Volume integral: not required This is the easiest way to do the problem: disk/washer method. But you could solve the problem using the shell methods and two integrals:
b) Write the integral that gives the volume generated when the same region is revolved around the line You do not need to evaluate the integral.
Volume integral: , this simplification not required (Harder, with the shell method: ) (The numeric answer, both ways, is , not required ) 22. The region bounded by , , and is revolved around the -axis.
a) Using the method of cylindrical shells, write the integral that gives the volume generated. Put your integral in the box below.
Volume integral:
, this simplification not required
b) Evaluate this integral. Put your final numeric answer in the box below.