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Exam 15: Vector Anal

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Exam 1: Preparation for Calculus125 Questionsadd course
Exam 2: Limits and Their Properties85 Questionsadd course
Exam 3: Differentiation193 Questionsadd course
Exam 4: Applications of Differentiation154 Questionsadd course
Exam 5: Integration184 Questionsadd course
Exam 6: Differential Equations93 Questionsadd course
Exam 7: Applications of Integration119 Questionsadd course
Exam 8: Integration Techniques and Improper Integrals130 Questionsadd course
Exam 9: Infinite Series181 Questionsadd course
Exam 10: Conics, Parametric Equations, and Polar Coordinates114 Questionsadd course
Exam 11: Vectors and the Geometry of Space130 Questionsadd course
Exam 12: Vector-Valued Functions85 Questionsadd course
Exam 13: Functions of Several Variables173 Questionsadd course
Exam 14: Multiple Integration143 Questionsadd course
Exam 15: Vector Anal142 Questionsadd course
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A tractor engine has a steel component with a circular base modeled by the vector-valued function A tractor engine has a steel component with a circular base modeled by the vector-valued function . Its height is given by . (All measurements of the component are given in centimeters.) Find the lateral surface area of the component. Round your answer to two decimal places. ​ . Its height is given by A tractor engine has a steel component with a circular base modeled by the vector-valued function . Its height is given by . (All measurements of the component are given in centimeters.) Find the lateral surface area of the component. Round your answer to two decimal places. ​ . (All measurements of the component are given in centimeters.) Find the lateral surface area of the component. Round your answer to two decimal places. ​

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Question 21

Identify the surface by eliminating the parameters from the vector-valued function Identify the surface by eliminating the parameters from the vector-valued function . ​ . ​

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Question 22

Find the curl of the vector field Find the curl of the vector field . ​ . ​

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Question 23

Evaluate Evaluate , where and S is the closed surface of the solid bounded by the graphs, and , and the coordinate planes. ​ , where Evaluate , where and S is the closed surface of the solid bounded by the graphs, and , and the coordinate planes. ​ and S is the closed surface of the solid bounded by the graphs, Evaluate , where and S is the closed surface of the solid bounded by the graphs, and , and the coordinate planes. ​ and Evaluate , where and S is the closed surface of the solid bounded by the graphs, and , and the coordinate planes. ​ , and the coordinate planes. ​

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Question 24

Let Let and let S be the cube bounded by the planes and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. ​ and let S be the cube bounded by the planes Let and let S be the cube bounded by the planes and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. ​ and Let and let S be the cube bounded by the planes and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. ​ . Verify the Divergence Theorem by evaluating Let and let S be the cube bounded by the planes and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. ​ as a surface integral and as a triple integral. ​

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Question 25

Find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. ​ Find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. ​ ​ ​

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Question 26

Set up and evaluate a line integral to find the area of the region R bounded by the graph of Set up and evaluate a line integral to find the area of the region R bounded by the graph of . .

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Question 27

Determine whether or not the vector field is conservative. ​ Determine whether or not the vector field is conservative. ​ ​ ​

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Question 28

Determine whether the vector field is conservative. If it is, find a potential function for the vector field. Determine whether the vector field is conservative. If it is, find a potential function for the vector field.

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Question 29

Find the value of the line integral Find the value of the line integral . . Find the value of the line integral . Find the value of the line integral .

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Question 30

Find the maximum value of Find the maximum value of where is any closed curve in the xy-plane, oriented counterclockwise. where Find the maximum value of where is any closed curve in the xy-plane, oriented counterclockwise. is any closed curve in the xy-plane, oriented counterclockwise.

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Question 31

Find the work done by a person weighing Find the work done by a person weighing pounds walking exactly one revolution up a circular helical staircase of radius feet if the person rises feet. pounds walking exactly one revolution up a circular helical staircase of radius Find the work done by a person weighing pounds walking exactly one revolution up a circular helical staircase of radius feet if the person rises feet. feet if the person rises Find the work done by a person weighing pounds walking exactly one revolution up a circular helical staircase of radius feet if the person rises feet. feet.

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Question 32

Use Stokes's Theorem to evaluate Use Stokes's Theorem to evaluate where and S is the first-octant portion of over . Use a computer algebra system to verify your result. ​ where Use Stokes's Theorem to evaluate where and S is the first-octant portion of over . Use a computer algebra system to verify your result. ​ and S is the first-octant portion of Use Stokes's Theorem to evaluate where and S is the first-octant portion of over . Use a computer algebra system to verify your result. ​ over Use Stokes's Theorem to evaluate where and S is the first-octant portion of over . Use a computer algebra system to verify your result. ​ . Use a computer algebra system to verify your result. ​

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Question 33

Evaluate Evaluate where S is the closed surface of the solid bounded by the graphs of and . ​​ ​ where S is the closed surface of the solid bounded by the graphs of Evaluate where S is the closed surface of the solid bounded by the graphs of and . ​​ ​ and Evaluate where S is the closed surface of the solid bounded by the graphs of and . ​​ ​ . ​​ Evaluate where S is the closed surface of the solid bounded by the graphs of and . ​​ ​ ​

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Question 34

Find the curl for the vector field at the given point. Find the curl for the vector field at the given point. , , Find the curl for the vector field at the given point. ,

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Question 35

Find the value of the line integral Find the value of the line integral . . Find the value of the line integral . Find the value of the line integral .

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Question 36

Find a piecewise smooth parametrization of the path C given in the following graph. ​ Find a piecewise smooth parametrization of the path C given in the following graph. ​

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Question 37

Let Let and let C be the triangle with vertices of oriented counterclockwise. Use Stokes's Theorem to evaluate . ​ and let C be the triangle with vertices of Let and let C be the triangle with vertices of oriented counterclockwise. Use Stokes's Theorem to evaluate . ​ oriented counterclockwise. Use Stokes's Theorem to evaluate Let and let C be the triangle with vertices of oriented counterclockwise. Use Stokes's Theorem to evaluate . ​ . ​

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Question 38

Find the flux Find the flux of through S, , where is the upward unit normal vector to S. ​ ​ , first octant ​ of through S, Find the flux of through S, , where is the upward unit normal vector to S. ​ ​ , first octant ​ , where Find the flux of through S, , where is the upward unit normal vector to S. ​ ​ , first octant ​ is the upward unit normal vector to S. ​ Find the flux of through S, , where is the upward unit normal vector to S. ​ ​ , first octant ​ ​ Find the flux of through S, , where is the upward unit normal vector to S. ​ ​ , first octant ​ , first octant ​

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Question 39

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. ​ Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. ​ ​ C: circle clockwise from to ​ ​ C: circle Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. ​ ​ C: circle clockwise from to ​ clockwise from Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. ​ ​ C: circle clockwise from to ​ to Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. ​ ​ C: circle clockwise from to ​ ​

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Question 40
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