MATH 224D – Spring 2024
Midterm 1
April 19, 2024
Name:
Student ID:
• You will have 50 minutes to complete the exam. Do not begin until instructed to do so.
• Please verify that your exam contains 4 questions spread over 6 pages.
• You are allowed to use a TI-30X IIS calculator and one double-sided, handwritten, 8.5” ×
11” note sheet for reference. You may not use a phone, smartwatch, earbuds, etc..
• All answers must be exact. Decimal approximations may not receive full credit.
• You must show your work on all problems. A correct answer without justification will likely
receive no credit.
• Please write your answers on the answer lines below each question.
• Please avoid writing within 0.5” of the edge of any page.
• If you need more room for your work, you may use the scratch work page, but please clearly
indicate that part of your answer is found there or it will not be graded.
• Any student found engaging in academic misconduct will receive a score of zero on this
exam.
Question Points Score
1 12
2 8
3 13
4 12
Total: 45
1
This page is for scratch work. If you want this work graded as part of your answer to a question,
please clearly indicate this on the page where that question appears.
Page 2
1. (12 points) Let D be the lamina in the xy-plane bounded by y = 2x2 and y = x2 + 1 with
constant density K. Find the mass m and the center of mass ( x̄, ȳ) of D.
m= ( x̄, ȳ) =
Page 3
2. (8 points) Let D be the region in the xy-plane defined by D = {( x, y) | 3 ≤ x2 + y2 ≤ 8},
and let S be the surface in R3 defined by the equation z = xy + 7. Find the surface area of the
portion of S that lies above D.
Surface area =
Page 4
p
3. (13 points) Let E be the solid bounded from below by the cone z = x2 + y2 and from above
by the sphere x2 + y2 + z2 = 2z. (Note: this sphere is not centered at the origin.) Suppose that E
has constant density K. Let Iz denote the moment of inertia of E about the z-axis.
(a) Set up but do not solve an integral to find Iz in Cartesian coordinates. (Hint: the projection
of E onto the xy-plane is a disk.)
Iz =
(b) Set up but do not solve an integral to find Iz in spherical coordinates. Your answer should
not mention x, y, or z.
Iz =
Page 5
4. (12 points) Let S be the unit square in the first quadrant of the uv-plane, i.e., S is bounded by
u = 0, u = 1, v = 0, and v = 1. Consider the coordinate transformation defined by
(
x = u + 2v
y = 2u + v2 .
Let R denote the image of S under the transformation.
∂( x,y)
(a) Compute the Jacobian ∂(u,v) of the transformation and use it to compute the area of R.
Your answer should not use part (b) below.
Area of R =
(b) Draw R in the xy-plane as accurately as possible. Clearly label the vertices of R, as well
as the functions (in the form y = f ( x )) that describe each of its four boundary curves.
(Hint: find the image of the vertices of S first; then the boundary curves of S must be sent to
boundary curves of R joining the corresponding vertices.)
Page 6
