Math 21D Practice Midterm Exam: Integration and Multivariable Calculus, Exams of Vector Analysis

A practice midterm exam for math 21d, focusing on integration and multivariable calculus. Students are required to sketch regions of integration, perform reversed integrals, integrate functions using trigonometric substitutions, find volumes of solid regions, and convert integrals to cylindrical coordinates. Problems include evaluating integrals with given limits and finding the mass of an apple using spherical coordinates.

Typology: Exams

Pre 2010

Uploaded on 07/30/2009

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Math 21D Name:
practice midterm exam Student ID#
February 2008 Section number
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO
FILL IN ABOVE INFORMATION (your name, etc) NOW!!
Show your work on every problem. Correct answers with no support work will not
receive full credit. Be organized and use the notation appropiately. No calculators
are allowed, nor is any assistance from classmates, notes, or books. You should only
have a writing and an erasing implement on your desk. No cell phones please.
Please write legibly!!
# Student’s Score Maximum possible Score
1 7
2 7
3 7
4 7
5 5
Total points 35
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Download Math 21D Practice Midterm Exam: Integration and Multivariable Calculus and more Exams Vector Analysis in PDF only on Docsity!

Math 21D Name: practice midterm exam Student ID# February 2008 Section number

DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO

FILL IN ABOVE INFORMATION (your name, etc) NOW!!

Show your work on every problem. Correct answers with no support work will not

receive full credit. Be organized and use the notation appropiately. No calculators

are allowed, nor is any assistance from classmates, notes, or books. You should only

have a writing and an erasing implement on your desk. No cell phones please.

Please write legibly!!

# Student’s Score Maximum possible Score

Total points 35

  1. Sketch the region of integration of the following integral and write an equivalent integral in the reversed order of integration. Then evaluate the new integral ∫ (^1)

0

∫ (^) x

x^2

xdydx.

  1. Integrate the function f (x, y) = 1/(1+x^2 +y^2 )^2 over the region enclosed by the triangle with vertices (0, 0), (1, 0), (1,
  1. Hint: Use a trigonometric substitution to solve the final integral x = tan(u).
  1. Find the volume of the region between the saddle z = xy and the cone z^2 = x^2 + y^2 and above the portion of the disk whose boundary is x^2 + y^2 = 1 that lies in the first quadrant.
  2. Convert to cylindrical coordinates and evaluate:

∫ (^1)

− 1

∫ √ 1 −x 2

−√ 1 −x^2

∫ (^) (x (^2) +y (^2) )

−(x^2 +y^2 )

21 xy^2 dzdydx

  1. The surface of an apple can be described in spherical coordinates by the equation ρ = 1 − cos(φ) (a revolution figure of half a cardiod). Apples have constant density 1, find the mass of the apple.