Calculus II (For Physical Sciences and Engineering)
Calculus 8th Edition by James Stewart
Dates: Jan 22, 2025 - April 28, 2025 on Mondays, Wednesdays, Fridays
This is an undergraduate course.
Name: Professor Joe Cutrone
Lecture Room: Maryland 110, Johns Hopkins University
Time: 10:00 - 11:50 am
Prof. Cutrone's Office: Krieger Hall 301
Prof. Cutrone's Office Hours: Mondays 11-12pm, Wednesdays 11-12pm, also by appointment.
Email Address: jcutron2 [at] jhu [dot] edu
Name: Professor Mee Seong Im
Lecture Room: Hodson 210, Johns Hopkins University
Time: 3:00 - 3:50 pm
Prof. Im's Office: Krieger Hall 419
Prof. Im's Office Hours: Mondays 4-5pm, Wednesdays 4-5pm, also by appointment. If I am not in my office, then I am still in Hodson 210 (our lecture room).
Email Address: meeseong [at] jhu [dot] edu
Head Teaching Assistant (Head TA): Emily Stroud
Teaching Assistants (TAs): Mitch Majure, Junu Park, Zhexing Zhang
TA Office Hours:
Emily Stroud: estroud2 [at] jhu [dot] edu
Mitch Majure: Mondays 2:00 - 4:00pm on Math Help room or Krieger 211.
Junu Park: Saturdays 7:00 - 9:00pm in Gilman 119.
Zhexing Zhang: Thursdays 5:00 - 7:00pm on Math Help room.
Class Assistants (CAs): Tyler Wunder, Cathy Wang, Mark Qu, Kashvi Ganesh, Isabella Iregui
Section Meeting Days and Times.
AS.110.109 (01): Tuesdays 8:00 - 8:50 am, with Zhexing Zhang (TA), Tyler Wunder (CA), with Professor Joe Cutrone
AS.110.109 (02): Tuesdays 6:00 - 6:50 pm, with Mitch Majure (TA), Cathy Wang (CA), with Professor Joe Cutrone
AS.110.109 (03): Thursdays 1:30 - 2:20 pm, with Mitch Majure (TA), Mark Qu (CA), with Professor Joe Cutrone
AS.110.109 (04): Tuesdays 3:00 - 3:50 pm, with Zhexing Zhang (TA), Kashvi Ganesh (CA), with Professor Mee Seong Im
AS.110.109 (06): Thursdays 4:30 - 5:20 pm, with Junu Park (TA), Isabella Iregui (CA), with Professor Mee Seong Im
Breakdown of the Grades (tentative):
In-person quiz: 10% (done in your section 5x or 6x, written by the Head TA, will be announced well in advance, 30 mins, closed notes and closed book)
Homework: 10% (lowest grade will be dropped, online, receive feedback immediately, 3 attempts, no timer, notes and books allowed, due Sundays at 23:59 ET)
Quizzes: 15% (lowest grade will be dropped, online, receive feedback immediately, 2 attempts, randomized questions, timer at 30 minutes, notes and books allowed, due Fridays at 23:59 ET)
First midterm: 20% (Friday, in-class)
Second midterm: 20% (Friday, in-class)
Final exam: 25% (in-class, this may replace a lower midterm grade)
Grading Scale.
A: 90 - 100
B: 80 - 89
C: 70 - 79
D: 63 - 69
F: < 63
"+" and "-" will be determined at the end of the semester.
There will absolutely be no make-up homework, quizzes, in-class quizzes, or exams.
It is your responsibility to check Canvas on a regular basis for your weekly homework and quiz assignments.
Athletics: For athletes with out-of-town events, we can email the exam to your coach. The coach then prints, proctors, and returns the exam to us via email. Please talk to your coach and have that coach reach out to us.
Note: if your Exam 1 and Exam 2 grades are higher than your Final Exam grade, your grades will remain the same. That is, your Exam 1 and Exam 2 grades will not replace your Final Exam grade.
Course Topics.
Techniques of Integration
7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitution
7.4 Integration of Rational Functions by Partial Fractions
Differential Equation
9.1 Modeling with Differential Equations
9.2 Direction Fields and Euler’s Method
9.3 Separable Equations
9.4 Models for Population Growth
9.5 Linear Equations
Parametric Equations and Polar Coordinates [no longer in the course syllabus]
10.1 Curves Defined by Parametric Equations
10.2 Calculus of parametric Curves
10.3 Polar Coordinates
10.4 Areas and Lengths in Polar Coordinates
Improper Integrals
7.8 Improper Integrals
Sequences and Their Limits
11.1 Sequences
Infinite Series and Convergence
11.2 Series
11.3 The Integral test and Estimates of Sums
11.4 The Comparison Tests
11.5 Alternating Series, Power Series, Radius of Convergence
11.6 Absolute Convergence and the Ratio and Root Tests
11.7 Strategies for Testing Series
11.8 Power Series
Calculus with Power Series, Taylor Series and Polynomials, Remainders
11.9 Representing Function as Power Series
11.10 Taylor and Maclaurin Series
11.11 Applications of Taylor Polynomials
Module 0: Review Sections 5.1 - 5.3 on your own, Prerequisites (Chapter 5)
Use Riemann Sums to approximate areas under a curve.
Evaluate the definite integral as a limit of Riemann Sums to compute exact areas under curves.
Apply the Fundamental Theorem of Calculus to evaluate definite integrals of continuous functions.
Day 0 (for those students who did not complete Calculus I in the past 2 semesters)
Module 1: Sections 5.4, 5.5, Review of Calculus I (Chapter 5)
Evaluate definite integrals using the Fundamental Theorem of Calculus.
Use indefinite integrals to find antiderivatives.
Apply the technique of u-substitution to find antiderivatives.
Jan 22 (Wed):
Go through Syllabus, Canvas
Section 5.4: Review of Calculus I: definite and indefinite integrals
Jan 24 (Fri):
Section 5.5: u-substitution
Module 2: Sections 6.1 - 6.3, Using integrals to find areas and volumes (Chapter 6)
Use integrals to find areas between curves.
Use integrals to find volumes of rotations of curves by the washer method and cylindrical shells.
Jan 27 (Mon):
Section 6.1: Areas between curves (integrate with respect to dx and dy)
Jan 29 (Wed):
Section 6.2: Volumes of solids of revolution
Jan 31 (Fri):
Section 6.2: Volumes of solids of revolution about x = a or y = c
Section 6.2: Volumes using cross-sectional area
Section 6.3: Volumes by cylindrical shells
An extension has been given on this week's homework and quiz (since we are slightly behind). Check Canvas for more detail.
Module 3: Sections 6.4, 6.5, 7.1, Work, average value, and integration by parts (Chapter 6)
Determine the work done by a function over an interval.
Calculate the average value of a function over an interval.
Find the integral of functions using integration by parts.
Feb 3 (Mon):
Section 6.3: Volumes by cylindrical shells (continued) (HW and online quiz have been extended, now due tonight 11:59 pm)
Section 6.4: Work, W = ΔKE = F d, Hooke's law
We will not cover work done by water pressure.
NOT ON EXAMS: You'll see online quiz and online HW questions about work, but Section 6.4 will not be on in-person quizzes or any exam, including the Final Exam.
Feb 5 (Wed):
Section 6.4: Work (one more example)
Section 6.5: Average value
Section 7.1: Integration by parts
Feb 7 (Fri):
Section 7.1: Integration by parts
Section 7.2: Integrals using powers of trigonometric functions (powers of sine and cosine)
Module 4: Sections 7.2 - 7.4, Techniques of integration (Chapter 7)
Learn further techniques to solve integrals using powers of trigonometric functions.
Learn when to use trigonometric substitutions to solve integrals.
Use the method of partial fractions to solve integrals of rational functions.
Feb 10 (Mon):
Section 7.2: Summarize powers of sine and cosine
Section 7.2: Powers of secant and tangent
Feb 12 (Wed):
Section 7.2: Summarize powers of secant and tangent
Section 7.3: Integrals using trigonometric substitution
Feb 14 (Fri):
Section 7.3: Integrals using trigonometric substitution (do 2 more examples)
Section 7.4: Integration of rational functions
Module 5: Sections 7.5 - 7.8, Numerical integration and improper integrals (Chapter 7)
Continue to practice techniques of integration in preparation for our first exam.
Use Tables of Integration or technology to help solve or approximate more difficult integrals.
Recognize improper integrals and how to solve them.
Feb 17 (Mon):
Section 7.4: Integration of rational functions (do 3 more examples)
Section 7.5: Techniques of integration
Feb 19 (Wed):
Section 7.6: Use Tables of Integration or technology to solve difficult integrals
Section 7.7: Midpoint Rule, Trapezoidal Rule, Simpson's Rule
NOT ON EXAMS: We will cover Sections 7.6 and 7.7 Numerical Integration in class, but Sections 7.6 and 7.7 will not be on any exam, including the Final Exam.
Feb 21 (Fri):
Section 7.7: Midpoint Rule, Trapezoidal Rule, Simpson's Rule (do 1 example)
Section 7.8: Improper integrals of Type I
Section 7.8 Improper Integrals will be on the exam.
Module 6: Review Chapters 5, 6, 7, no quiz or homework this week.
Feb 24 (Mon):
Section 7.8: Improper integrals of Type II
Section 7.8: Comparison test for improper integrals
Feb 26 (Wed):
Section 7.5: Techniques of integration (complete it)
Review for the Exam
Feb 28 (Fri):
Exam 1
50 minutes.
8 pages total, which includes the cover page.
7 problems; some problems have multiple parts.
No technology, calculators, books, notes, etc.
One problem is T/F (do not justify but clearly write True or False).
If you want us to grade the back of some of the pages, clearly draw an arrow (← or →) at the bottom of the problem, pointing towards your additional work.
Work through the problems quickly.
Show work for partial credit.
Put a box around your final solution.
If you have time left, go back and double check your work.
Bring your JHU Photo ID. Write your JHU ID on the cover sheet. This will be checked when you hand in your exam.
Module 7: Sections 11.1 - 11.4, Sequences and series (Chapter 11)
Define and find limits of sequences, if they exist.
Understand that a series is a sequence of partial sums.
Use tests to determine if a series convergences, and in specific situations, find what the series converges to.
Mar 3 (Mon):
Section 11.1: Sequences
Mar 5 (Wed):
Section 11.1: Sequences (continued)
Section 11.2: Series (geometric, telescoping, harmonic), divergence test
Mar 7 (Fri):
Section 11.2: Series (geometric, telescoping, harmonic), divergence test (continued)
Exam 1 Grades have been posted on Canvas. You have until Monday March 10, 5pm, to ask for a regrade.
Exam 1 statistics will be posted after the grades are finalized.
Yes, there will be a curve.
Module 8: Sections 11.5 - 11.7, Tests for series convergence (Chapter 11)
Finish the study of tests for convergence for series by studying the Alternating Series Test, Ratio, and Root Test.
Understand the difference between absolute convergence and conditional divergence.
Work on learning when to use each test to test the convergence and divergence of a series.
Mar 10 (Mon):
Section 11.3: Integral test
Section 11.4: Comparison test (we are slightly behind; HW and online quiz extended until tonight)
NOT ON EXAMS: Section 11.3 Remainder Estimate for integral test
The highest grade is 92/100.
We have given each student +8 bonus points.
Professor Cutrone: 5 out of 88 are A's.
Professor Im: 7 out of 43 are A's.
Mar 12 (Wed):
Section 11.5: Alternating series test and Absolute convergence
Mar 14 (Fri):
Section 11.6: Ratio and Root test
Spring Break:
Mar 17 (Mon): Spring Break
Mar 19 (Wed): Spring Break
Mar 21 (Fri): Spring Break
Module 9: Sections 11.8 - 11.10, Taylor and power series (Chapter 11) [We will spend 2 weeks on Taylor and power series]
Work with Power Series; in particular, finding their interval and radius of convergence.
Find Taylor Series and Maclaurin Series to find higher order derivatives and evaluate limits.
Represent functions as power series by manipulating, both algebraically and using calculus, the geometric series.
Mar 24 (Mon): [Substitute: Dr. Haihan Wu]
Section 11.8: Power series
Mar 26 (Wed): [Substitute: Dr. Haihan Wu]
Section 11.8: Power series (continued)
Section 11.9: Representations of functions as power series
Mar 28 (Fri): [Substitute: Dr. Haihan Wu]
Section 11.9: Representations of functions as power series (continued)
Module 10: Sections 11.8 - 11.10, Taylor and power series (Chapter 11) [We will spend 2 weeks on Taylor and power series]
Work with Power Series; in particular, finding their interval and radius of convergence.
Find Taylor Series and Maclaurin Series to find higher order derivatives and evaluate limits.
Represent functions as power series by manipulating, both algebraically and using calculus, the geometric series.
Mar 31 (Mon):
Section 11.8: Power series (review)
Section 11.9: Representations of functions as power series (review)
Section 11.10: Taylor and Maclaurin series
Apr 2 (Wed):
Section 11.10: Taylor and Maclaurin series (continued)
Apr 4 (Fri):
Section 11.10: Taylor and Maclaurin series (continued)
Section 11.7: Strategy for testing series
NOT ON EXAMS: Section 11.10 Binomial Series will be discussed in class, but they will not be on the exams.
NOT ON EXAMS: Section 11.10 Remainder Estimates, Taylor's Inequality
NOT ON EXAMS: Section 11.11 Applications of Taylor Polynomials
Module 11: Review Chapter 11, no quiz or homework this week.
Apr 7 (Mon): [Substitute: Dr. Haihan Wu]
Review for Exam 2: what to study? Exam 2 will cover only Chapter 11 material.
Summary of Chapter 11 Flow Chart
Exam 2 Practice Problems with solutions
11.1 Limit of sequences, recurrence relation
11.2 Series (geometric, telescoping, harmonic 1/n), divergence test, recurrence relation-- compute the limit
11.3 Integral test
11.4 Comparison test
11.4 Limit comparison test
11.5 Alternating series test
11.5 Absolutely convergent and conditionally convergent
11.6 Ratio test
11.6 Root test
11.7 Learn when to use each test
11.8 Power series
11.9 Representations of functions as power series
11.10 Taylor and Maclaurin series
Apr 9 (Wed): [Substitute: Dr. Haihan Wu]
Review for the Exam
Apr 11 (Fri):
Exam 2
50 minutes.
8 pages total, which includes the cover page.
7 problems; some problems have multiple parts.
No technology, calculators, books, notes, etc.
One problem is T/F (do not justify but clearly write True or False).
If you want us to grade the back of some of the pages, clearly draw an arrow (← or →) at the bottom of the problem, pointing towards your additional work.
Work through the problems quickly.
Show work for partial credit.
Put a box around your final solution.
If you have time left, go back and double check your work.
Bring your JHU Photo ID. Write your JHU ID on the cover sheet. This will be checked when you hand in your exam.
Module 12: Sections 8.1, 8.3, 8.5, Applications of integration (Chapter 8)
Understand and apply the formula for arclength.
Apply integration to problems in physics, specifically to find moments and centers of mass.
Use integration to identify distributions of continuous random variables, and use these distributions to find probabilities.
Apr 14 (Mon):
Section 8.1: Arc length
Apr 16 (Wed):
Section 8.3: Applications to Physics and Engineering
NOT ON THE FINAL EXAM: Section 8.3 Applications to Physics and Engineering
Apr 18 (Fri):
Section 8.3: Short review of moments of inertia and center of mass
Section 8.5: Probability
The highest grade is 99/100.
We have given each student +0 bonus points.
Professor Cutrone: 5 out of 88 are A's.
Professor Im: 7 out of 43 are A's.
Module 13: Sections 9.1 - 9.3, Differential equations (Chapter 9)
Define and recognize the order of a differential equation.
Sketch solutions to differential equations using direction fields.
Use integration to solve separable equations.
Apr 21 (Mon):
Section 8.5: Probability (continued: median for exp distribution and normal distribution)
Section 9.1: Modeling with Differential Equations
Apr 23 (Wed):
Section 9.1: Modeling with Differential Equations (continued: the rate of population growth is proportional to the population)
Section 9.2: Direction Fields and Euler's Method
Section 9.3: Separable Equations
Apr 25 (Fri):
Review for the Final Exam
Practice Final Exams
Practice Final Exam 1 Solutions
Module 14: Review, no quiz or homework this week.
Apr 28 (Mon):
Review for the Final Exam
May 9 (Fri, 9am - 12pm, Gilman 50):
Final Exam
Bring a photo ID.
Bring plenty of pencils or pens.
There are 9 questions, and most questions have multiple parts.
No calculators, technology, books, or notes.
Every other seat should be EMPTY.
Show work for partial credit.
9 - 10 am: Joe Cutrone, Tyler Wunder, Emily Stroud
10 - 11 am: Tyler Wunder, Emily Stroud, Mee Seong Im
11 - 12 pm: Tyler Wunder, Emily Stroud, Mee Seong Im
Final Exam
The highest grade is 98/100.
23.14% of the students received a B or above.
Good job, everyone! Have a great summer!
May 9 (Fri, 12pm - ?, Krieger 413):
Grading Party! TAs and CAs, stop by for some food:
Sandwiches and wraps, potato salads, chips and salsa, fruits, vegetables, desserts, drinks, etc.