MA 2005 A-95 Sample Exam 3 Name here

Show your work in the space provided. Unsupported answers may not receive full credit. Use the backs of the pages if needed. (In the real exam, there will be plenty of space for your work.)

1. Set up and evaluate a triple integral in cylindrical coordinates to compute the mass of a body contained inside the cylinder for if the density is given by .

2. Set up and evaluate a triple integral in spherical coordinates to find the volume of the portion of the unit sphere in the octant defined by the inequalities , , and .

3. If

compute the divergence, , and the curl, .

4. Show that the following identity is true for all sufficiently differentiable scalar functions and two-dimensional vector fields .

Expand the derivatives using the chain rule. Then compute the two terms on the right hand side of the identity and compare.

5. Compute the line integral

where and C is the straight line y= -3x+2 for .

Answer: The curve can be parametrized by , for . Then

and

so

6. Compute the line integral

where C is the helix for and the vector field is given by

and

Next, compute the dot product

So we have

7. Suppose that is a potential function for the two dimensional vector field and that C is smooth curve in the plane that starts at point at t=0 and ends at point at t=1. Show that

Answer: Since is a potential for , we know that

Now we write our parametric curve as

So we have

And

Note that the integrand is the same as

so we have

The final step is to realize that , , , and , which gives the desired result.

8. Set up, but do not evaluate a double integral that will give the surface area of the surface defined by z=xy for and .

Answer: First, we need to compute dS. The easiest way is to use the formula

which gives

So the area is given by

9. Use the divergence theorem to evaluate

where the vector field is given by

and the surface S is the unit cube , , and , and is the outward unit normal to the surface.

Answer: The divergence theorem says that

where R is the region bounded by the surface S. We have

which gives us

10. For the vector field and surface S in the previous problem, compute directly the surface integral

where is the single face of the unit cube defined by z=1, , and .

Answer: For this face of the cube, we have z=1, , and . So we have

and

11. Challenge question: Why are the answers to the two previous problems the same?