MATH 215 FALL 2025
Homework Set 8: §15.6 - 15.9
Only some of the questions on this and other homework sets will be graded.
Due November 3, no later than 11:59pm, submitted through Gradescope.
You may work on these problems in groups (in fact, this is encouraged!), but you must submit your own set of solutions. Please neatly show your work!
Question 1: Find the mass and center of mass of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and 3x + 4y + 2z = 12, if the density of the region is given by f(x, y, z) = 2 + x.
Question 2: Let E be the region between the cylinders x2 + y2 = 1 and x2 + y2 = 16, above the xy-plane, and below the plane z = y + 4. Evaluate
Question 3: Take a sphere of radius R with mass density proportional to the square of the distance from the origin in such a way that the maximum density is 9. Cut this sphere with two planes that intersect along a diameter at an angle of π/3. (This shape should look roughly like the segment of an orange.) Find the mass of this wedge of the sphere.
Question 4:
(a) Sketch the region of integration for the integral
Rewrite this integral as an equivalent iterated integral with integration order dy dx dz.
(b) Sketch the region of integration for the integral
Rewrite this integral as an equivalent iterated integral with integration order dx dy dz. (This one is a bit more challenging than the first part.)
Question 5: (a) Let E be the region in the first quadrant that is above the line y = x/3, below the line y = 3x, and between the curves defined by xy = 3 and xy = 27. Sketch the region E and evaluate
(Hint: Try u = xy and v = y/x.) (Another Hint: This problem is suspiciously similar to Question 6 from the Fall 2023 Midterm 2, in case you need inspiration.)
(b) Find where f (x, y) = 3y2 - 3xy - 4x2 and R is the quadrilateral with vertices (0, 2), (3, 0), (5, 4), and (2, 6). Hint: There may be a straightforward but tedious way to solve this problem as well as a faster, more subtle, way to solve this problem. (∗cough∗ see part (a) ∗cough∗)
Question 6: A massive body E of constant density of mass equal to one generates a gravitational potential at a point (0, 0, a) given by
(We have set the gravitational constant equal to one.) In this problem we take E to be a solid ball of radius R centered at the origin.
(a) Compute V (0, 0, a), assuming that a > R) (i.e. the point is outside the ball).
(b) Compute V (0, 0, a), assuming that 0 ≤ a < R (i.e. the point is inside the ball). Hint: You are going to need to compute two integrals in this case.
Extra Credit: Although Fubini’s theorem holds for most functions met in practice, we must still exercise some caution. This exercise gives a function for which this theorem fails. Let R be the unit square (i.e. if (x, y) ∈ R then 0 ≤ x ≤ 1, 0 ≤ y ≤ 1), and consider the function
(a) Using the substitution y = x tan θ, compute the integral
(b) Using the substitution x = y tan θ, compute the integral
(c) Are your answers to parts (a) and (b) the same? Should they be the same? If the answers are the same, explain why. If the answers are diferent, explain why this does not violate Fubini’s theorem.