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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 .

    Answer: Write . Then

    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



  6. Compute the line integral

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



    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


    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


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

    Answer: You are on your own with this one.

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William W. Farr
Thu Oct 5 09:54:11 EDT 1995