MATH 1D Foothill College Multivariable Integral Calculus Questions

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Mathematics

math 1D

Foothill College

MATH

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Multivariable Integral Calculus, Multivariable Integral Calculus Multivariable Integral Calculus Multivariable Integral Calculus Multivariable Integral Calculus Multivariable Integral Calculus Multivariable Integral Calculus Multivariable Integral Calculus

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Page 1 of 6 Page 2 of 6 1. (2 points) Let a thin plate be bounded by π¦ = β4 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 donβt include all the limits needed, you donβt get credits. (A)-(D) Express (DO NOT calculate) an integral to denote the total mass of the plate in the following coordinates. (A) ππ₯ππ¦ in rectangular coordinates (solution) (B) ππ¦ππ₯ in rectangular coordinates (solution) (C) ππππ in polar coordinates (solution) (D) ππππ 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) Page 3 of 6 2. (2 points) Let a cylindrical solid bounded by both π₯2 + π¦2 = 1 and π§ = β4 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 donβt include all the limits needed, you donβt 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 donβt get credits. (A) ππ¦ππ₯ in rectangular coordinates (solution) (B) ππππ in polar coordinates (solution) (C) ππ§ππ¦ππ₯ in rectangular coordinates (solution) (D) ππ§ππππ in cylindrical coordinates (solution) (E) ππππππ in spherical coordinates (solution) (F) Evaluate the volume using one of the coordinates from (A) to (E) above. Simplify your answer. (solution) Page 4 of 6 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§πsin(π₯ π ), cos(ππ₯ 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) (B) β Γ β(π₯π (solution) cos π¦ π¦2020π§πsin(π₯ π π¦π§ ) π¦π§ + cos(ππ₯ 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§πsin(π₯ π π¦π§ ) π¦π§ , cos(ππ₯ 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) Page 5 of 6 4. (5 points) According to Coulombβs Law, the electrostatic field πΈβ at the point π due to a charge π at the origin is π given by πΈβ (π) = π . Note that div πΈβ = 0 and curl πΈβ = β0. ||π ||3 You must show your work clearly and use concepts you have learned in Math1D. If you donβt include all the limits needed, you donβt get credits. (A) Find the domain of πΈβ . (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 πΆ on the π¦π§plane given below, not using a parameterization of πΆ, but a parameterization of your own curve πΆβ. 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, β1 β€ π₯ β€ 3, β1 β€ π¦ β€ 5 on the π₯π¦-plane, oriented counterclockwise when viewed from the positive z-axis. (solution) Page 6 of 6 (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 πΈβ 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, β1 β€ π₯ β€ 2, β3 β€ π¦ β€ 4, β5 β€ π§ β€ 6, oriented inward. (solution) Page 7 of 6
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