Problem 1. (a) Set up a triple integral (using ) that represents the volume of the solid region bounded by the sphere and the cone .
(b) Change variables for your integral in part (a) so that you can compute the volume of .
Problem 2. For this problem we will use a function that depends on three positive, constant numbers :
(a) What is ? What is and what is ? Describe in words and/or pictures what does to the unit sphere (centered at the origin).
(b) Compute the derivative matrix of and compute the determinant of this derivative matrix.
(c) Compute the volume of the region bounded by the ellipsoid given by the equation
Problem 3. Evaluate the (improper, one-dimensional) integral
Problem 4. Consider the helix curve that is parameterized by the function where .
(a) Compute the length of the curve .
(b) If the function measures the density at every point on , compute the total mass of .
(c) Compute the amount of work done by the vector field in moving a particle along .
Problem 5. Let be any differentiable function. Prove / compute that
Problem 6. (a) Compute the line integral
where and is the ellipse (traversed in a counter-clockwise direction) given by
(b) Compute the line integral
where is the same vector field from part (a), but is only the top half of the ellipse from part (a) (also given a counter-clockwise orientation).
Problem 7. State the following integral theorems:
The Gradient Theorem, Greens’ Theorem, Stokes’ Theorem, the Divergence Theorem.
Problem 8. Use Greens Theorem to compute the area of the region given by
Problem 9. Set up a double integral that represents the surface area of the graph for over the domain .
Problem 10. (a) Write down the meaning of the following symbols:
(b) Which of the above symbols are used for line integrals? Which are used for surface integrals?
(c) Given a surface , write down an integral expression that represents its surface area (assuming the surface has been parameterized by a function ). In addition, write down an expression for a unit normal to .
Problem 11. Compute the flux of the vector field out of the unit sphere .
Problem 12. (a) Compute the flux of the vector field out of the cylindrical surface
(b) Draw a picture of the surface . Is this surface the boundary of a solid region? Why or why not?
(c) Describe / draw a picture of the boundary of or explain why it has no boundary. If does have a boundary, how should it be oriented so as to be consistent with the outward normal ?
(d) Compute the flux
Problem 13. Suppose is a solid region that has volume . If is the boundary surface of , compute the flux of out of the surface where