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Michigan Math 116 Calculus II: Topics, Study Guide and Practice

A study guide for University of Michigan Math 116 (Calculus II): integration techniques, applications, infinite series and Taylor series, with practice.

University of Michigan Updated 3 min read

At a glance

Course code
Math 116
Credits
4 credits (no credit with Math 156, 186, 215, 216, 276, 285 or 286)
Offered
Typically every term: Fall, Winter, Spring and Summer
Prerequisites
Math 115 or AP credit
Covers
Techniques of integration, applications of integration, infinite series and Taylor series
Next course
Math 215 (Calculus III: Multivariable and Vector Calculus)
Practice here
Calculus 2
On this page
  1. What Michigan Math 116 covers
  2. Math 116 topic map
  3. Math 116 vs its honors alternatives
  4. Example problems at the Math 116 level
  5. How to do well in Michigan Math 116

Michigan Math 116 is Calculus II at the University of Michigan, the middle course in the standard sequence of Math 115, Math 116 and Math 215. Students take it after Math 115 or with AP credit, and it is the course where integration becomes a toolkit rather than a single rule. This guide maps every Math 116 topic to free practice with instant feedback.

What Michigan Math 116 covers

According to the official course page, Math 116 presents integral calculus from four points of view: graphs, tables, formulas and verbal descriptions. Its topics are techniques of integration, applications of integration, Taylor series and infinite series. The classroom is interactive, with both individual and team homework.

That lines up closely with a standard Calculus 2 course. You extend antiderivatives with substitution, parts and partial fractions, use integrals to compute quantities such as volume and arc length, and then spend the last part of the term on series, including approximating functions with Taylor polynomials.

Math 116 topic map

Math 116 topic Practice Read
Substitution U-substitution refresher u-substitution, u-substitution rule
Integration by parts Integration by parts Integration by parts
Partial fractions and trig substitution Partial fractions Partial fractions, trig substitution
Choosing a technique Choosing a strategy Antiderivatives
Improper integrals Integrating to infinity Improper integrals
Volume and area Volumes with washers Volumes of revolution, area between curves
Arc length, work and mass Arc length and surface area Definite integral properties
Sequences and series Series and partial sums Geometric series, convergence tests
Power series Power series and radius Power series, ratio test
Taylor series Taylor polynomials Taylor series, Taylor's theorem

Math 116 vs its honors alternatives

Course Approach
Math 116 (Calculus II) Conceptual, four points of view, individual and team homework
Math 186 (Honors Analytic Geometry and Calculus II) Somewhat more theoretical, much of the same material
Math 156 (Applied Honors Calculus II) Uses MAPLE, emphasizes science and engineering applications

After Math 116 the natural sequel is Math 215, and a student who has done very well can enter the honors sequence with Math 285.

Need a refresher on derivatives first? Start with the Michigan Math 115 guide. When you finish, the Math 215 guide covers multivariable calculus.

Example problems at the Math 116 level

1. Integration by parts. Evaluate \int x\cos x\,dx with u = x and dv = \cos x\,dx:

\int x\cos x\,dx = x\sin x - \int \sin x\,dx = x\sin x + \cos x + C

2. A convergent improper integral.

\int_0^{\infty} e^{-2x}\,dx = \lim_{t \to \infty}\left[-\frac{1}{2}e^{-2x}\right]_0^t = \lim_{t \to \infty}\left(\frac{1}{2} - \frac{1}{2}e^{-2t}\right) = \frac{1}{2}

3. Estimating with a Taylor polynomial. The degree 3 Taylor polynomial of e^x at 0 is 1 + x + \frac{x^2}{2} + \frac{x^3}{6}, so

e^{0.1} \approx 1 + 0.1 + 0.005 + 0.000167 = 1.105167

which matches the true value 1.105171\ldots to five decimal places.

How to do well in Michigan Math 116

  • Drill the choice, not just the technique. Mixed sets like choosing a strategy train you to recognize when parts, substitution or partial fractions fits.
  • Learn a short list of series by heart. The geometric series and the Maclaurin series for e^x, \sin x and \cos x appear constantly. Practice them in common Maclaurin series.
  • Set up applications carefully. For volume and work problems, sketch a slice and write its size before integrating.
  • Keep the formula sheet nearby. The calculus formula sheet collects integrals and series on one page.
  • Test yourself with practice exams, and review common calculus mistakes before each midterm.

Tip: almost every convergence argument rests on the geometric series formula. Master it before the other tests.

Practice it now

U-substitution refresher

Solve real problems on this topic with instant feedback, hints and a step-by-step walkthrough for every answer. Free, no sign-up needed to start.

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Frequently asked questions

What is Math 116 at the University of Michigan?
Math 116 is Calculus II, a 4-credit course on integral calculus. It covers techniques of integration, applications of integration, infinite series and Taylor series, and it is usually offered in every term.

What do I need before Math 116?
The listed prerequisite is Math 115 or AP credit. You should be comfortable with derivatives, antiderivatives and the Fundamental Theorem of Calculus before starting.

What comes after Math 116?
Math 215 (Calculus III: Multivariable and Vector Calculus) is the natural sequel. A student who has done very well in Math 116 could enter the honors sequence with Math 285.

Is Michigan Math 116 hard?
Many students find Calculus II the hardest course in the sequence, because integration techniques and series tests require choosing a method rather than following one recipe. Practicing mixed problems where you must pick the method yourself is the best preparation.

What is the difference between Math 116, Math 156 and Math 186?
All three cover much of the same material. Math 186 is a somewhat more theoretical honors course, while Math 156 uses MAPLE and emphasizes applications to science and engineering.

Written by the PracticeCalculus team

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Related university courses

Michigan Math 115 Calculus 1 Calculus I A study guide for University of Michigan Math 115 (Calculus I): functions, derivatives, applications and a first look at integrals, with practice. Michigan Math 215 Calculus 3 Calculus III: Multivariable and Vector Calculus A study guide for University of Michigan Math 215 (Calculus III): vectors, partial derivatives, multiple integrals and vector calculus, with practice.