Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

MTH 254 Fall Term 2007 Test 3: Double and Triple Integrals, Exams of Calculus

Portland Community CollegeCalculus

The instructions and problems for a university-level mathematics test focusing on double and triple integrals. The test includes questions on finding areas and volumes of regions and solids using double and triple integrals, as well as reversing the order of integration. The document also includes examples of students' incorrect reasoning and the correct solutions.

Typology: Exams

Pre 2010

Uploaded on 08/16/2009

koofers-user-tbr
koofers-user-tbr 🇺🇸

8 documents

1 / 10

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Page 1 of 10
MTH 254, Fall Term 2007
Test 3 – Due December 13, 2007 Name
1. Use double integrals to find the area of the region shaded in Figure A.
The boundary curve of the region is
(
)
2cos 1yx
π
=
−. Mark up
Figure B with all of the labels (and color coding) discussed and
illustrated in class. The integrals should be evaluated on your
calculator.
Figure B
Figure A
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download MTH 254 Fall Term 2007 Test 3: Double and Triple Integrals and more Exams Calculus in PDF only on Docsity!

MTH 254, Fall Term 2007

Test 3 – Due December 13, 2007 Name

  1. Use double integrals to find the area of the region shaded in Figure A.

The boundary curve of the region is y = 2 cos (^) ( π x )− 1. Mark up Figure B with all of the labels (and color coding) discussed and illustrated in class. The integrals should be evaluated on your calculator.

Figure B

Figure A

2. The top of the solid shown in Figure X is the surface z = y 3/ 2^ + 1. The lateral sides of the

solid are the planes x = 0 and y = 4 along with the parabolic cylinder y = x^2.

Find, without the use of your calculator , the volume of the solid; use a double integral in your

calculation. Please note … in order to have success, you need to make the proper choice for your order of integration. Include a sketch of the region of integration including all of the labels (and color coding) discussed and illustrated in class.

Page 3 has been left blank to finish this problem. Please set the integral up on page 2 and show the integration on page

  1. Make sure that there is clear evidence on your paper that you actually did evaluate the integral by hand; antiderivative formulae that appear out of nowhere will not be presumed to have come from your pure thoughts. Arithmetic calculations that would amaze Newton himself were they actually done in your head need to be somewhat worked out on your paper.

Figure X

  1. Mr. Hedgewart gave a test over double integrals. Mr. Hedgewart asked his class to reverse the

order of integration for the double integral

2 2

(^1 2 )

x

∫ − ∫ − x x^ −^ x y^ +^ y^ + dy dx.^ All questions in

this problem are about this task.

a. Sketch out the region of integration. Do not worry about fancy labels … just roughly sketch out the region (with axes labels and scales) and actually shade the region of integration.

b. While taking the test, Leo decided he would exploit the symmetry of the region of integration. He drew a mighty fine picture; he had his boundary curves dutifully labeled in a color coded fashion and his axis limits actually labeled as points on the axes. Leo even asserted his answer affirmatively; that is, Leo wrote as his final statement the equation 2 2

(^1 2 3 1 12 )

x

∫ − ∫ − x x^ −^ x y^ +^ y^ +^ dy dx^ =^ ∫ ∫ yx^ −^ x y^ +^ y^ + dx dy.^ Despite all this, when

Leo got his paper back he had lost more than half the points for the problem. Write a sentence or two that explains the error in Leo’s reasoning – that is, explain why those two double integrals are not equal. A picture might help with your explanation.

c. After the test, Gemini and Orion were talking about the problem. Orion was saying what a pain it was that he had to use 4 double integrals when he reversed the order of integration. Gemini noted that she was able to do it with only 3 double integrals. Both students had valid solutions to the problem. I’ll give you full credit if you can come up with Gemini’s solution, and ¾ credit if you come up with Orion’s solution. There are a couple of caveats,

though. You may only submit one of the two solutions for credit and to receive all of the

credit you need to have a fully labeled picture (or pictures) of the region of integration that supports your integrals. Work this part of the problem on page 5.

5. The top of the solid shown in Figure S is the plane z = y and the

bottom is the plane z = 0. The lateral sides of the solid are the planes

x = 0 and y = 4 along with the parabolic cylinder y = x^2. Find the

volume of the solid using a triple integral of form G

∫∫∫ dx dy dz.^ Mark

up Figure T with all of the labels (and color coding) discussed and

illustrated in class. Evaluate the resultant triple integral on your

calculator.

Figure T

Figure S

6. On page 9, a square region in the xy -plane has been subdivided into 6 sub-regions; the 6 sub

regions have been identified by the Roman numerals I – VI. The interior boundaries of the

region are the line y = x , the circle x^2^ + y^2 = 1 , and the parabola y = x^2.

Each of the double integrals below finds the areas of one of these regions or a combination of two or more of the regions. In each provided blank, state the region(s) whose area is being found by the attendant double integral. The answers for the first two double integrals have been given to help you understand what I’m asking for. No work should be shown other than your answer. Feel free to mark up Figure G in any way you see fit.

1 1 0 0

∫ ∫ dx dy finds the area of region(s)^ I - VI^.

/4 1 0 0

r dr d

π

∫ ∫ θ finds the area of region(s)^ IV and V^.

1/ 2 1 2 0

x x

dy dx

−

∫ ∫ finds the area of region(s)^.

( ) ( )

/4 sec( ) 0 sec tan

r dr d

π θ θ θ

∫ ∫ θ finds the area of region(s)^.

2

1 0

x

∫ ∫ x dy dx finds the area of region(s)^.

2

1 1

∫ ∫ 0 1 − y dx dy finds the area of region(s)^.

/4 sec ( (^) ) tan( )

0 0 r dr d

π θ θ

∫ ∫ θ finds the area of region(s)^.

  1. When I introduced the concept of rectangular integrals I estimated the volume of the solid in

Figure I by embedding rectangular parallelepipeds between the xy -plane and the surface at the

top ( z = x y − 3 y + 25 ). Specifically, I broke up the triangular region in the xy -plane into 6

rectangles and used the midpoint of each rectangle to determine the height of the parallelepiped.

In class we first partitioned the y -axis into 3 intervals of equal width and

then partitioned the embedded rectangles into 2 rectangles of equal

length (parallel to the x -axis). In this problem you need to replicate

that process except you need to first partition the x -axis into 2

intervals of equal width and then partition each of the embedded

rectangles into 3 rectangles of equal length (parallel to the y -axis).

Organize your work in a manner similar to what we showed in page 2 of that document. You do not need to show the formal Riemann sum – you can just add up the numbers in the appropriate column and state an appropriate conclusion.

Figure I