Deck 17: Line and Surface Integrals

Question 1

Define where Compute the vector assigned to the point by the vector field

Answer 1

Question 2

Let be the curve .

A) Parametrize using polar coordinates.
B) Find the length of

Answer 2

A) B) 8

Question 3

Define At what points is normal to the vector

Answer 3

Question 4

Let . Express in terms of the unit radial vector in .

Question 5

Compute the vector assigned to the point by the vector field

Question 6

Let be a differentiable function of r, and let .
Express in terms of the unit radial vector .

Question 7

Find a potential function for the field by inspection.

Question 8

Calculate the work performed by the force field on a particle that moves in the counterclockwise direction around the quarter circle . Assume that is in Newtons and the unit of distance is the meter.

Question 9

Find a potential function for by inspection.

Question 10

Compute the work performed in moving a particle along the path by the force .

Question 11

Define and Compute

Question 12

Define Compute the vector assigned to the point by the vector field

Question 13

Compute , where is the part of the ellipse joining the point to the point and .

Question 14

Let . Express in terms of the unit radial vector in .

Question 15

Let . Express in terms of the unit radial vector in .

Question 16

Let be the triangle with vertices at the points and in counterclockwise order.
Compute the line integral .

Question 17

Determine whether the vector field is a conservative vector field, and if so, find a potential function for by inspection.

Question 18

Compute , where is the part of the parabola starting at and ending at .

Question 19

Let denote the closed curve of intersection of the hemisphere and the cylinder oriented counterclockwise.
Compute where .

Question 20

Evaluate , where is the rectangle in with vertices at , and , in this order.

Question 21

Let .

A) Determine whether is conservative, and if so, find a potential function for
B) Compute the integral where C is the path from to .

Question 22

Let . Find a function so that is conservative in and for all .

Question 23

Evaluate for the path C shown in the following figure. The path consists of the line segment from (1, 0, 0) to (0, 1, 0), followed by the circular arc in the yz-plane with radius 1 joining (0, 1, 0) to (0, 0, 1).

Question 24

Find where is the path from to and .

Question 25

Compute the work performed in moving a particle along the path by the force

Question 26

Compute the work performed in moving a particle along the path by the force

Question 27

Compute the length of the parametric curve ,

Question 28

Compute the line integral where C is the segment from to and .

Question 29

Let .
Determine whether is conservative. If so, find a potential function.

Question 30

Compute where C is the quarter circle on the plane starting at and ending at

Question 31

Evaluate where C is the parametric curve from to .

Question 32

Compute the mass of the curve if the mass density is and are measured in centimeters.

Question 33

Let . Compute where is the segment joining the points and together with the quarter circle centered at the origin and joining the points and . The integration is performed from to .

Question 34

Let

A) Compute
B) Evaluate where C is the parametric curve from to

Question 35

Let C be the curve , , , , and let .
The value of is which of the following?

A)
B)
C)
D)
E) None of the answers is correct.

Question 36

Compute the line integral where C is the line segment from to and .

Question 37

Compute the mass of the curve if the mass density is

Question 38

Compute where and is the path from to .

Question 39

Evaluate where C is the line segment joining the points and

Question 40

Compute the work performed in moving a particle along the path by the force

Question 41

Find where is the path from to and

Question 42

Let .
Compute where C is the spiral

Question 43

Let S denote the part of the surface inside the cylinder The area of S is closest to which of the following?

A)
B)
C)
D)
E)

Question 44

In the paraboloid , the charge density is equal to the distance from the plane. Find the total charge in the paraboloid.

Question 45

Let .

A) Determine whether is conservative, and if so, find a potential function for
B) Compute the line integral where C is the path consisting of: : the parabola from to : the line segment from to and : the semicircle in the plane with diameter where and

Question 46

Compute along the curve shown in the following figure.

Question 47

Consider the vector field Find the formula for so that is conservative and

Question 48

Consider the vector field Find the formula for so that is conservative and

Question 49

Let .

A) Does have a potential function? If so, find it.
B) Find the integral where C is the semicircle oriented counterclockwise with the diameter , where and .

Question 50

Let .

A) Determine whether has a potential function, and if so, find a potential function for .
B) Compute where , is the curve , from to and is the straight line from to

Question 51

Compute where S is the part of the plane which lies inside the half cylinder .

Question 52

Find where is the path from to and .

Question 53

Find where is the path from to and

Question 54

Consider the vector field Find the formula for so that is conservative and

Question 55

Compute the surface integral for the surface .

Question 56

The surface integral where is the part of the plane which lies inside the cylinder is which of the following?

A)
B)
C)
D) 0
E) None of the answers is correct.

Question 57

The mass of the part of the cone with and density is which of the following?

A)
B)
C)
D)
E) None of the answers is correct.

Question 58

Find the surface area of the surface .

Question 59

Evaluate the surface integral where S is the part of the cone between the planes and .

Question 60

Find where is the path from to and

Question 61

Let S be the surface .
The area of S is approximately which of the following

A)
B)
C)
D)
E) None of the answers is correct.

Question 62

The integral where and with normals pointing to the positive y direction is equal to which of the following?

A)
B)
C)
D)
E)

Question 63

Compute where S is the part of the plane in the first octant bounded by the planes and

Question 64

Evaluate where is the portion of the paraboloid contained within the cylinder

Question 65

Evaluate where S is the surface cut from the paraboloid by the planes , and .

Question 66

Evaluate where and is the surface oriented outward.

Question 67

Evaluate where is the portion of the cylinder bounded between the planes and

Question 68

Find the flux of across the surface , oriented outward.

Question 69

Compute where S is the part of the surface between the planes and

Question 70

Compute the flux of water through the parabolic cylinder (oriented in the positive y-direction)
S : , , where the velocity vector is (in ).

Question 71

Compute the area of the surface .

Question 72

Find the surface area of the portion of the sphere satisfying

Question 73

Let where and . Let be a sector of angle in the sphere of radius , centered at the origin and oriented outward. Find .

Question 74

The area of the surface is which of the following?

A)
B)
C)
D)
E) None of the answers is correct.

Question 75

Find the flux of the field through the surface which is the part of the paraboloid where .

Question 76

Compute the surface integral where and is the surface of the sphere centered at the origin with radius 4.

Question 77

Find the surface area of the portion of the cone within the cylinder

Question 78

Find the flux of the field through the surface oriented upward.

Question 79

Parametrize the surface first using Cartesian coordinates and then using cylindrical coordinates. Assume x and y take on all real values.

Question 80

Find the area of the ellipse cut from the plane by the cylinder .