Calculus 3

Math 3C (21303) Final Exam Sections 15.1 to 16.8 Show all work!

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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 ∫ ∫ 7𝑦 𝑑𝑥 𝑑𝑦 !"#$%!

& # $# 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 – y2. Set up only integrals that would give the volume of the region with the following orders: (a) ∫ ∫ ∫ 𝑑𝑧 𝑑𝑦 𝑑𝑥

(b) ∫ ∫ ∫ 𝑑𝑥 𝑑𝑦 𝑑𝑧

4) The region in this problem is like the region in Gertrude’s problem. The edges of the parallelogram

are the lines 𝑦 = ' " 𝑥 + 4; 𝑦 = '

" 𝑥 − 1; 𝑦 = −𝑥 + 3; 𝑦 = −𝑥 + 5. Use the change of variables

𝑢 = 𝑦 − ' " 𝑥; 𝑣 = 𝑦 + 𝑥 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: 𝑥 = "

( 𝑣 − "

( 𝑢; 𝑦 = "

( 𝑢 + '

( 𝑣.

22 𝑥" + 2𝑦" 𝑑𝑥𝑑𝑦 )

=

5) Evaluate ∫ (𝑥 + 𝑥𝑦 + 𝑧)𝑑𝑠* along the curve C parameterized by 𝑟(𝑡) = ⟨𝑡,2𝑡,3 − 2𝑡⟩; 0 ≤ 𝑡 ≤ 5

R

6) By any legal method, find the circulation of the field �⃗�(𝑥,𝑦,𝑧) = ⟨2𝑥,7𝑧,7𝑦⟩ around the circle parameterized by �⃗�(𝑡) = ⟨cos(𝑡) ,sin(𝑡) ,0⟩;0 ≤ 𝑡 ≤ 2𝜋

7) Evaluate ∫ �⃗� ∙ 𝑑𝑟* where �⃗�(𝑥,𝑦) = ⟨𝑦

",𝑥⟩ and C is the counterclockwise path around the perimeter of the rectangle 0 ≤ 𝑥 ≤ 2, 0 ≤ 𝑦 ≤ 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 �⃗�(𝑥,𝑦,𝑧) = ⟨5𝑥,2𝑦,3𝑧⟩ 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 – Theo Rem.]