calc_tutorial_17_3_013

 
Problem Statement

{5}
{2}
pow(2,2)
3*5
15/5
32*5/5

Use the Divergence Theorem to evaluate the flux .

F = <5x3, 0, 5z3>, S is the octant of the sphere x2 + y2 + z2 = 4, in the first octant x ≥ 0, y ≥ 0, z ≥ 0.

 
Step 1

Question Sequence

Question 1

Recall how to find the divergence of a vector field F = <F1, F2, F3>.

div(F) = (∂F1 / ∂x) + (∂F2 / ∂y) + (∂F3 / ∂z)

Recall the Divergence Theorem. Let S be a closed surface that encloses a region W in R3. Assume that S is piecewise smooth and is oriented by normal vectors pointing to the outside of W. Let F be a vector field whose domain contains W. Then we have the following equation.

To evaluate the flux, we will need to find the divergence of the vector field F and then use the appropriate coordinates and bounds of integration to integrate the divergence. Calculate div(F) for F = <5x3, 0, 5z3>.

div(F) = x2 + z2

Substitute the expression for div(F) into the equation from the Divergence Theorem.

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 2

Since we need to compute the triple integral over the first octant of the sphere x2 + y2 + z2 = 4, we convert to spherical coordinates.

We convert using the standard spherical conversions.

A.
B.
C.

Restricting the sphere to the first octant places the following bounds on ρ, θ, and φ.

0 ≤ ρ ≤

A.
B.
C.

0 ≤ θ ≤

A.
B.
C.

0 ≤ φ ≤

A.
B.
C.

Convert the integrand to spherical coordinates in terms of sin θ and sin φ. Recall the Pythagorean identity sin2x + cos2x = 1.

15x2 + 15z2 = 15(ρ cos θ sin φ)2 + 15(ρ cos φ)2

= 15ρ2(cos2θ sin2φ + cos2φ)

= 15ρ2(cos2θ sin2φ + ( − sin2φ))

= 15ρ2(−sin2φ( - cos2φ) + )

= 15ρ2( − sin2φ sin2θ)

Correct.
Incorrect.

 
Step 3

Question Sequence

Question 3

Use information from Part 2 to write the appropriate triple integral. Recall that for spherical coordinates, the differential dV becomes ρ2 sin φ dρ dφ dθ.

=

A.
B.
C.
Correct.
Incorrect.

 
Step 4

Question Sequence

Question 4

Evaluate the flux by computing the triple integral.

= π

Correct.
Incorrect.