Math 224E - Spring 2023
Midterm 1
April 21, 2023
Name:
Student ID Number:
• There are 5 questions on this exam across 7 pages. Please make sure that your exam contains 5
questions and 7 pages.
• You will have 50 minutes to complete the exam. Do not begin until instructed to do so.
• You are allowed to use a TI-30X calculator and one hand-written, two-sided, 8.5′′ × 11′′ sheet of notes.
• All answers must be exact. A decimal approximation for an exact answer will not receive full
credit. Answers should be reasonably simplified.
• You must show your work on all problems. A correct answer without justification will likely receive no
credit.
• Please box your final answers.
• If you need more room for your work, you may use the back of the sheet or the additional sheet at the
end of the exam, but please indicate clearly where the remainder of your work can be found.
• Any student found engaging in academic misconduct will receive a score of 0 on this exam.
Question Points Score
1 10
2 10
3 10
4 10
5 10
Total: 50
1
1. (10 points) Let D be the region bounded by the parabola y = x2 and y = 4. Compute
¨
x2 y + xy 2 dA
D
2
2. (10 points) Let D = {(x, y) | x ≥ 0, x2 + y 2 ≤ 4} be the (right) half-disk of radius 2, and let S be the
surface defined by y 2 − x2 + 1. Compute the surface area of S above D.
3
3. (10 points) Let S be the ball of radius R with density proportional to the square of the distance to the
center of the ball (with proportionality constant K). Compute the mass of S, as a function of R and K.
4
4. (10 points) Let E be the region bounded by the paraboloids z = 9 − x2 − y 2 and z = x2 + y 2 − 9 with
constant density σ. Find the moment of inertia of E around the z-axis, in terms of σ.
5
n o
x2 y2
5. (10 points) Let E be the ellipse given by E = (x, y) | A2 + B 2 ≤ 1 . Use the change of variables (r, θ)
given by the rule
x = Ar cos(θ) y = Br sin(θ)
to compute the area of E. Remember that the determinant of a 2 × 2 matrix is given by
a b
det = ad − bc
c d
x2 y2
and that in this change of coordinates, A2 + B2 = 1 when r = 1.
6
Use this page for additional scratch paper.
