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1. (2 points) Let a thin plate be bounded by π¦ = β4 β π₯2, π¦ = 1, π¦ = 2, and π₯ = 2. Assume that the mass density
of the plate is the distance from the π¦-axis.
You must show your work clearly and use concepts you have learned in Math1D. If you donβt include all
the limits needed, you donβt get credits.
(A)-(D) Express (DO NOT calculate) an integral to denote the total mass of the plate in the following
coordinates.
(A) ππ₯ππ¦ in rectangular coordinates
(solution)
(B) ππ¦ππ₯ in rectangular coordinates
(solution)
(C) ππππ in polar coordinates
(solution)
(D) ππππ in polar coordinates
(solution)
(E) Express (DO NOT calculate) an integral to denote the average mass density of the plate in your choice of
coordinates.
(solution)
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2. (2 points) Let a cylindrical solid bounded by both π₯2 + π¦2 = 1 and π§ = β4 β π₯2 β π¦2 on the π₯π¦-plane.
You must show your work clearly and use concepts you have learned in Math1D. If you donβt include all
the limits needed, you donβt get credits.
(A)-(E) Express (DO NOT calculate) an integral to denote the volume of the solid in the following coordinates.
If you split the solid unnecessarily, you donβt get credits.
(A) ππ¦ππ₯ in rectangular coordinates
(solution)
(B) ππππ in polar coordinates
(solution)
(C) ππ§ππ¦ππ₯ in rectangular coordinates
(solution)
(D) ππ§ππππ in cylindrical coordinates
(solution)
(E) ππππππ in spherical coordinates
(solution)
(F) Evaluate the volume using one of the coordinates from (A) to (E) above. Simplify your answer.
(solution)
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3. (1 point) Determine whether each of the following is defined or not. Explain why or why not. Find it if it is
defined, and explain how.
You must show your work clearly and use concepts you have learned in Math1D.
(A) β β β Γ< π₯π
cos π¦
π¦π§
π¦π§
π¦2020π§πsin(π₯ π ), cos(ππ₯ π π¦3π§) β π₯π§, ππ₯
2π§ cos( π₯π¦π§)
π¦ + π§2020 >.
(i) Is it well-defined? Explain briefly why or why not.
(solution)
(ii) If it is defined, find it and explain briefly how.
(solution)
(B) β Γ β(π₯π
(solution)
cos π¦
π¦2020π§πsin(π₯ π
π¦π§ )
π¦π§
+ cos(ππ₯ π π¦3π§) β π₯π§ + π π₯
2π§
cos( π₯π¦π§)
π¦ + π§2020).
(i) Is it well-defined? Explain briefly why or why not.
(solution)
(ii) If it is defined, find it and explain briefly how.
(solution)
(C) β β β< π₯π
(solution)
cos π¦
π¦2020π§πsin(π₯ π
π¦π§ )
π¦π§
, cos(ππ₯ π π¦3π§) β π₯π§, π π₯
2π§
cos( π₯π¦π§)
π¦ + π§2020 >.
(i) Is it well-defined? Explain briefly why or why not.
(solution)
(ii) If it is defined, find it and explain briefly how.
(solution)
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4. (5 points) According to Coulombβs Law, the electrostatic field πΈβ at the point π due to a charge π at the origin is
π
given by πΈβ (π) = π . Note that div πΈβ = 0 and curl πΈβ = β0.
||π ||3
You must show your work clearly and use concepts you have learned in Math1D. If you donβt include all
the limits needed, you donβt get credits.
(A) Find the domain of πΈβ .
(solution)
(B) Does the Curl Test apply? Explain why or why not.
(solution)
(C) Determine whether the vector field is path-independent or not. Explain why.
(solution)
(D) Express (DO NOT compute) a line integral to find work done by the vector field along a curve πΆ on the π¦π§plane given below, not using a parameterization of πΆ, but a parameterization of your own curve πΆβ. Your
description must be clear including all the vectors and limits.
(solution)
(E) Use the Fundamental Theorem of Calculus for Line Integral to find work done by the vector field along
the boundary of the closed rectangle, β1 β€ π₯ β€ 3, β1 β€ π¦ β€ 5 on the π₯π¦-plane, oriented counterclockwise when
viewed from the positive z-axis.
(solution)
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(F) Can we use the Stokesβ Theorem to find the work done in (E) above? Explain why or why not. If yes, use the
theorem to find it.
(solution)
(G) Does the Divergence Test apply? Explain why or why not.
(solution)
(H) Let π be the sphere of radius 5 centered at the origin and oriented outward. Find the flux of πΈβ through S.
(solution)
(I) Can you evaluate the flux integral in (H) using the Divergence Theorem? Explain why or why not. If yes, find
it using the theorem.
(solution)
(J) Determine whether the vector field is a curl field or not. Explain why.
(solution)
(K) Without direct computations, find the flux of the vector field through the sphere oriented inward defined
by (π₯ β 5)2 + π¦2 + π§2 = 1.
(solution)
(L) Without direct computations, find the flux of the vector field through the boundary of the closed box,
β1 β€ π₯ β€ 2, β3 β€ π¦ β€ 4, β5 β€ π§ β€ 6, oriented inward.
(solution)
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