# 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.

Source: https://practicecalculus.com/courses/university-of-michigan-math-116
Updated: 2026-11-03

## 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](/learn/calculus-2)

## Key takeaways

- Math 116 is Calculus II at Michigan, the second course after Math 115 and before Math 215.
- It focuses on integral calculus: techniques of integration, applications of integration, infinite series and Taylor series.
- Like Math 115, it treats each idea through graphs, tables, formulas and words, with individual and team homework.
- Math 186 is the more theoretical honors version and Math 156 is an applied honors alternative for science and engineering.

**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](/learn/calculus-2/integration-techniques/u-substitution-refresher) | [u-substitution](/blog/u-substitution), [u-substitution rule](/rules/u-substitution-rule) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts) |
| Partial fractions and trig substitution | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions](/blog/partial-fractions), [trig substitution](/blog/trig-substitution) |
| Choosing a technique | [Choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) | [Antiderivatives](/blog/antiderivatives) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Volume and area | [Volumes with washers](/learn/calculus-1/applications-of-integrals/volumes-washers) | [Volumes of revolution](/blog/volumes-of-revolution), [area between curves](/blog/area-between-curves) |
| Arc length, work and mass | [Arc length and surface area](/learn/calculus-2/improper-integrals-and-applications/arc-length-and-surface-area) | [Definite integral properties](/rules/definite-integral-properties) |
| Sequences and series | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) | [Geometric series](/blog/geometric-series), [convergence tests](/blog/convergence-tests) |
| Power series | [Power series and radius](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series), [ratio test](/rules/ratio-test) |
| Taylor series | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-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](/courses/university-of-michigan-math-115). When you finish, the [Math 215 guide](/courses/university-of-michigan-math-215) 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](/learn/calculus-2/integration-techniques/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](/learn/calculus-2/power-series-parametric-polar/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](/formulas) collects integrals and series on one page.
- **Test yourself** with [practice exams](/practice), and review [common calculus mistakes](/blog/common-calculus-mistakes) before each midterm.

> **Tip:** almost every convergence argument rests on the [geometric series formula](/rules/geometric-series-formula). Master it before the other tests.

## 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.