Math
3C (21303) Final Exam
Sections 15.1 to 16.8
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1) Consider the shaded region shown below. The graphs are portions of x = -y2, and y = x + 12.
(a) Set up one iterated integral that would give the total area of this region. (7 points)
(b) Evaluate your iterated integral from part (a) (3 points)
2) Convert the Cartesian integral β« β« 7π¦ ππ₯ ππ¦
!”#$%!
&
#
$# into an equivalent polar integral. Do not
evaluate.
3) Consider the shaded region shown below. It is bounded by the coordinate planes, the plane
z + y = 10, and the parabolic cylinder x = 100 β y2. Set up only integrals that would give the
volume of the region with the following orders:
(a) β« β« β« ππ§ ππ¦ ππ₯
(b) β« β« β« ππ₯ ππ¦ ππ§
4) The region in this problem is like the region in Gertrudeβs problem. The edges of the parallelogram
are the lines π¦ = ‘
”
π₯ + 4; π¦ = ‘
”
π₯ β 1; π¦ = βπ₯ + 3; π¦ = βπ₯ + 5. Use the change of variables
π’ = π¦ β ‘
”
π₯; π£ = π¦ + π₯ to rewrite the integral in terms of u and v. Do not evaluate.
Here is some assistance for computing the new integrand and the Jacobian:
π₯ = ”
(
π£ β ”
(
π’; π¦ = ”
(
π’ + ‘
(
π£.
22 π₯” + 2𦔠ππ₯ππ¦
)
=
5) Evaluate β« (π₯ + π₯π¦ + π§)ππ * along the curve C parameterized by π(π‘) = β¨π‘,2π‘,3 β 2π‘β©; 0 β€ π‘ β€ 5
R
6) By any legal method, find the circulation of the field οΏ½βοΏ½(π₯,π¦,π§) = β¨2π₯,7π§,7π¦β© around the circle
parameterized by οΏ½βοΏ½(π‘) = β¨cos(π‘) ,sin(π‘) ,0β©;0 β€ π‘ β€ 2π
7) Evaluate β« οΏ½βοΏ½ β ππ* where οΏ½βοΏ½(π₯,π¦) = β¨π¦
“,π₯β© and C is the counterclockwise path around the
perimeter of the rectangle 0 β€ π₯ β€ 2, 0 β€ π¦ β€ 5. [Hint: A Green way to compute this would
involve less ink and paper, compared to computing directly.]
8) Determine the flux of the field οΏ½βοΏ½(π₯,π¦,π§) = β¨5π₯,2π¦,3π§β© out of a closed circular cylinder of radius 4,
parallel to and centered on the z-axis from z = -1 to z = 7. [Hint: The divergence of attention from a
task can be costly β Theo Rem.]