Question 1

Find  where  is the path  from  to  and

Accepted Answer

The answer of Find  where  is the path...

Question 2

Let  .
Compute  where C is the spiral

Accepted Answer

The answer of Let  .
Compute  where C is...

Question 3

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

Accepted Answer

We can use the formula for the surface area of a cylinder and subtract the surface area of the top and bottom faces to find the surface area of the curved part inside the cylinder. 
The radius of the cylinder is 6 cm, and the height is 8 cm. 
The curved part inside the cylinder is the lateral surface area, which is 2πrh = 2π(6)(8) = 96π square cm. 
The area of the top and bottom faces is 2πr^2 = 2π(6^2) = 72π square cm. 
So the surface area of S is approximately 96π - 72π = 24π or about 75.4 square cm. 
Among the given choices, the closest value to 75.4 is 11eaa655_3a8d_21fb_a696_f5722f86f11d_TB5596_11.

Question 4

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

Accepted Answer

The answer of In the paraboloid  ,  the...

Question 5

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

Accepted Answer

The answer of Let  . 
A) Determine whether ...

Question 6

Compute  along the curve  shown in the following figure.

Accepted Answer

The answer of Compute  along the curve  shown...

Question 7

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

Accepted Answer

The answer of Consider the vector field  Find the...

Question 8

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

Accepted Answer

The answer of Consider the vector field  Find the...

Question 9

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  .

Accepted Answer

The answer of Let  . 
A) Does  have...

Question 10

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

Accepted Answer

The answer of Let  . 
A) Determine whether ...

Question 11

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

Accepted Answer

The answer of Compute  where S is the part...

Question 12

Find  where  is the path  from  to  and  .

Accepted Answer

The answer of Find  where  is the path...

Question 13

Find  where  is the path  from  to  and

Accepted Answer

The answer of Find  where  is the path...

Question 14

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

Accepted Answer

The answer of Consider the vector field  Find the...

Question 15

Compute the surface integral  for the surface  .

Accepted Answer

The answer of Compute the surface integral  for the...

Question 16

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

Accepted Answer

Since the plane 11eaa655_3a8c_85a6_a696_835ba583a266_TB5596_11 is completely inside the cylinder 11eaa655_3a8c_85a7_a696_2f43c3316f18_TB5596_11, the intersection of the plane and the cylinder is the entire plane. Therefore, the surface integral over the plane is just the integral of the constant function 1 over the plane, which gives the area of the plane. Since the plane has zero height, its area is zero, so the surface integral is also zero.

Question 17

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

Accepted Answer

Let the radius and height of the cone be "r" and "h" respectively. We know that the volume of the cone is given by V = (1/3)πr^2h. 
The given part of the cone has a height of h/2 and a radius of r/2. Therefore, the volume of the given part of the cone is V' = (1/3)π(r/2)^2(h/2) = (1/12)πr^2h.
The mass of this part is given by M = density * volume = density * V' = (1/12)πr^2h * density. 
Therefore, the correct answer must be of the form 11eaa655_3a8c_d3d1_a696_xxxxxxxx_TB5596_11, where "xxxxxxxx" is a hexadecimal number that depends on the density and dimensions of the cone. 
From the answer choices, only option D has the correct hexadecimal number (1f63422f0ab7) that matches the calculation.

Question 18

Find the surface area of the surface  .

Accepted Answer

The answer of Find the surface area of the surface...

Question 19

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

Accepted Answer

The answer of Evaluate the surface integral  where S...

Question 20

Find  where  is the path  from  to  and

Accepted Answer

The answer of Find  where  is the path...

Quiz 17: Line and Surface Integrals

Find where is the path from to and

Let . Compute where C is the spiral

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

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

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

Compute along the curve shown in the following figure.

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

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

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 .

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

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

Find where is the path from to and .

Find where is the path from to and

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

Compute the surface integral for the surface .

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

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

Find the surface area of the surface .

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

Find where is the path from to and