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Prac 3 A

This document is a practice exam for 18.02 Multivariable Calculus from MIT OpenCourseWare. It contains 7 multi-part problems testing a variety of multivariable calculus concepts like finding centers of mass, moments of inertia, line integrals of vector fields, double integrals over regions, and applying Green's theorem.

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43 views2 pages

Prac 3 A

This document is a practice exam for 18.02 Multivariable Calculus from MIT OpenCourseWare. It contains 7 multi-part problems testing a variety of multivariable calculus concepts like finding centers of mass, moments of inertia, line integrals of vector fields, double integrals over regions, and applying Green's theorem.

Uploaded by

remino1
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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MIT OpenCourseWare

http://ocw.mit.edu

18.02 Multivariable Calculus


Fall 2007

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
18.02 Practice Exam 3 A

1. Let (x̄, ȳ) be the center of mass of the triangle with vertices at (−2, 0), (0, 1), (2, 0) and
uniform density δ = 1.
a) (10) Write an integral formula for ȳ. Do not evaluate the integral(s), but write explicitly the
integrand and limits of integration.
b) (5) Find x̄.

2. (15) Find the polar moment of inertia of the unit disk with density equal to the distance
from the y-axis.

3. Let F~ = (ax2 y + y 3 + 1)ı̂ + (2x3 + bxy 2 + 2)̂ be a vector field, where a and b are constants.

a) (5) Find the values of a and b for which F~ is conservative.

b) (5) For these values of a and b, find f (x, y) such that F~ = ∇f .

�
c) (5) Still using the values of a and b from part (a), compute F~ · d~r along the curve C such
C
that x = et cos t, y = et sin t, 0 ≤ t ≤ π.
�
4. (10) For F~ = yx3 ı̂ + y 2 ̂, find C F~ · d~r on the portion of the curve y = x2 from (0, 0) to (1, 1).

5. Consider the region R in the first quadrant bounded by the curves y = x2 , y = x2 /5, xy = 2,
and xy = 4.
a) (10) Compute dxdy in terms of dudv if u = x2 /y and v = xy.

b) (10) Find a double integral for the area of R in uv coordinates and evaluate it.

6. a) (5) Let C be� a simple closed curve going counterclockwise around a region R. Let
M = M (x, y). Express M dx as a double integral over R.
C
�
b) (5) Find M so that M dx is the mass of R with density δ(x, y) = (x + y)2 .
C

7. Consider the region R enclosed by the x-axis, x = 1 and y = x3 .


a) (5) Use the normal form of Green’s theorem to find the flux of F~ = (1 + y 2 )̂ out of R.
b) (5) Find the flux out of R through the two sides C1 (the horizontal segment) and C2 (the
vertical segment).
c) (5) Use parts (a) and (b) to find the flux out of the third side C3 .

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