Center for Academic Resources in Engineering (CARE)
Peer Exam Review Session
Math 241 − Calculus III
Midterm 3 Worksheet Solutions
The problems in this review are designed to help prepare you for your upcoming exam. Questions pertain
to material covered in the course and are intended to reflect the topics likely to appear in the exam. Keep
in mind that this worksheet was created by CARE tutors, and while it is thorough, it is not comprehensive.
In addition to exam review sessions, CARE also hosts regularly scheduled tutoring hours.
Tutors are available to answer questions, review problems, and help you feel prepared for your
exam during these times:
Session 1: Wed, 11/1, 5-7 pm - Jakob, Camila, Gabe @ CIF 4025
Session 2: Thurs, 11/2, 5-7 pm - Matthew, Rose @ CIF 2039
Can’t make it to a session? Here’s our schedule by course:
https://care.grainger.illinois.edu/tutoring/schedule-by-subject
Solutions will be available on our website after the last review session that we host.
Step-by-step login for exam review session:
1. Log into Queue @ Illinois: https://queue.illinois.edu/q/queue/845
2. Click “New Question”
3. Add your NetID and Name
4. Press “Add to Queue”
Please be sure to follow the above steps to add yourself to the Queue.
Good luck with your exam!
, Math 241 − Calculus III Midterm 3 Exam Review
1. Compute the double integral over the indicated rectangle. Confirm your answer by switching the
order of integration and recomputing.
ZZ
2x − 4y 3 dA R = [−5, 4] × [0, 3]
R
Z 3 Z 4 Z 4 Z 3
3
2x − 4y dxdy 2x − 4y 3 dydx
0 −5 −5 0
Z 3 Z 4
−9 − 36y 3 dy 6x − 81 dx
0 −5
3 4
− 9y − 9y 4 0
= −756 3x2 − 81x −5
= −756
2. Making an appropriate change of variables, compute the following double integral over the region
bound by a circle of radius 2 and a circle of radius 5.
ZZ
2 +y 2
ex dA
D
Using polar coordinates gives the following integral
ZZ Z 2π Z 5
x2 +y 2 2
e dA = rer drdθ
D 0 2
Then using a u−substitution u = r2 for the r dependence
2π 25
1
Z Z
eu dudθ = π(e25 − e4 ) ≈ 2.26 × 1011
2 0 4
2 of 9
Peer Exam Review Session
Math 241 − Calculus III
Midterm 3 Worksheet Solutions
The problems in this review are designed to help prepare you for your upcoming exam. Questions pertain
to material covered in the course and are intended to reflect the topics likely to appear in the exam. Keep
in mind that this worksheet was created by CARE tutors, and while it is thorough, it is not comprehensive.
In addition to exam review sessions, CARE also hosts regularly scheduled tutoring hours.
Tutors are available to answer questions, review problems, and help you feel prepared for your
exam during these times:
Session 1: Wed, 11/1, 5-7 pm - Jakob, Camila, Gabe @ CIF 4025
Session 2: Thurs, 11/2, 5-7 pm - Matthew, Rose @ CIF 2039
Can’t make it to a session? Here’s our schedule by course:
https://care.grainger.illinois.edu/tutoring/schedule-by-subject
Solutions will be available on our website after the last review session that we host.
Step-by-step login for exam review session:
1. Log into Queue @ Illinois: https://queue.illinois.edu/q/queue/845
2. Click “New Question”
3. Add your NetID and Name
4. Press “Add to Queue”
Please be sure to follow the above steps to add yourself to the Queue.
Good luck with your exam!
, Math 241 − Calculus III Midterm 3 Exam Review
1. Compute the double integral over the indicated rectangle. Confirm your answer by switching the
order of integration and recomputing.
ZZ
2x − 4y 3 dA R = [−5, 4] × [0, 3]
R
Z 3 Z 4 Z 4 Z 3
3
2x − 4y dxdy 2x − 4y 3 dydx
0 −5 −5 0
Z 3 Z 4
−9 − 36y 3 dy 6x − 81 dx
0 −5
3 4
− 9y − 9y 4 0
= −756 3x2 − 81x −5
= −756
2. Making an appropriate change of variables, compute the following double integral over the region
bound by a circle of radius 2 and a circle of radius 5.
ZZ
2 +y 2
ex dA
D
Using polar coordinates gives the following integral
ZZ Z 2π Z 5
x2 +y 2 2
e dA = rer drdθ
D 0 2
Then using a u−substitution u = r2 for the r dependence
2π 25
1
Z Z
eu dudθ = π(e25 − e4 ) ≈ 2.26 × 1011
2 0 4
2 of 9
