# Duke MATH 112L Laboratory Calculus II: Topics and Practice

> A study guide for Duke MATH 112L Laboratory Calculus II: integration methods, improper integrals, series, Taylor polynomials and differential equations.

Source: https://practicecalculus.com/courses/duke-math-112l
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 112L
- **Credits:** Not stated on the department course page
- **Offered:** Fall and/or spring
- **Prerequisites:** MATH 106L or 111L, or consent; not open to students with credit for MATH 122 or 122L
- **Covers:** Methods and applications of integration, functions defined by integration, improper integrals, probability and distributions, infinite series, Taylor polynomials, series solutions of differential equations, systems of differential equations and Fourier series
- **Next course:** MATH 212 (Multivariable Calculus) lists MATH 112L as a prerequisite
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- MATH 112L is the second semester of Duke's laboratory calculus sequence, with lab projects, group work and written reports.
- It covers integration methods and applications, improper integrals, series and Taylor polynomials.
- It adds topics many Calculus 2 courses leave out: probability distributions, systems of differential equations and Fourier series.
- Students with AP credit for MATH 21 usually take MATH 122L instead when it is offered.

**Duke MATH 112L**, Laboratory Calculus II, is the second semester of Duke's laboratory calculus sequence. Like [MATH 111L](/courses/duke-math-111l), it pairs calculus with lab projects, group work and written reports. Its topic list is wider than most Calculus 2 courses, reaching probability, systems of differential equations and Fourier series. This guide maps each topic to free practice with instant feedback.

## What MATH 112L covers

The Duke math department lists the topics as methods of integration, applications of integrals, functions defined by integration, improper integrals, an introduction to probability and distributions, infinite series, Taylor polynomials, series solutions of differential equations, systems of differential equations, and Fourier series.

The prerequisite is MATH 106L or 111L, or consent. The course is not open to students who have had MATH 122 or 122L, and the department notes that AP or placement credit for MATH 21 is not sufficient if MATH 122L is being offered. MATH 112L is typically offered in the fall and/or spring.

## MATH 112L topic map

| MATH 112L topic | Practice | Read |
| --- | --- | --- |
| Substitution | [U-substitution refresher](/learn/calculus-2/integration-techniques/u-substitution-refresher) | [U-substitution](/blog/u-substitution) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts guide](/blog/integration-by-parts) |
| Other methods of integration | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions guide](/blog/partial-fractions) |
| Applications of integrals | [Volumes with washers](/learn/calculus-1/applications-of-integrals/volumes-washers) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Functions defined by integration | [Accumulation functions](/learn/calculus-1/integrals/accumulation-functions) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1) |
| Improper integrals | [Infinite discontinuities](/learn/calculus-2/improper-integrals-and-applications/infinite-discontinuities) | [Improper integrals](/blog/improper-integrals) |
| Probability and distributions | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals practice](/practice-problems/improper-integrals) |
| Infinite series | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) | [Convergence tests](/blog/convergence-tests) |
| Convergence tests | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Ratio test](/rules/ratio-test) |
| Taylor polynomials | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series) |
| Series solutions of differential equations | [Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Power series](/blog/power-series) |
| Differential equations and modeling | [Logistic growth](/learn/calculus-2/improper-integrals-and-applications/logistic-growth) | [Differential equations practice](/practice-problems/differential-equations) |

Systems of differential equations and Fourier series are also on the list. They go beyond the lessons on this site, but both build on the integration, series and differential equations skills above.

## MATH 112L or MATH 122L?

Duke has two routes into the second semester. MATH 112L continues the laboratory sequence from MATH 106L or 111L. [MATH 122L](https://math.duke.edu/courses/introductory-calculus-ii-applications), Introductory Calculus II with Applications, is for students with advanced placement credit for MATH 21, and covers sequences and series, the definition of the integral and its uses, Taylor and Fourier series, and differential equations with weekly labs. You cannot take both, and either one can lead to MATH 212.

## Example problems at the MATH 112L level

**1. A probability density.** For $f(x) = 2e^{-2x}$ on $[0, \infty)$, the total probability is

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

**2. A Taylor polynomial.** The degree 3 Taylor polynomial of $\cos x$ at $0$ is $1 - \frac{x^2}{2}$, since the odd terms vanish.

**3. A series solution idea.** If $y' = y$ and $y = \sum a_n x^n$, matching coefficients gives $a_{n+1} = \frac{a_n}{n+1}$, so with $a_0 = 1$, $y = e^x$.

## How to do well in MATH 112L

- **Master integration methods early.** Every later topic, from probability to improper integrals, uses them. The [techniques practice](/learn/calculus-2/integration-techniques/techniques-practice) mixes them like an exam.
- **Connect improper integrals and probability.** A density must integrate to 1, which makes improper integrals feel concrete.
- **Treat series as a toolkit.** Taylor polynomials, series solutions and Fourier series all approximate functions with simpler pieces.
- **Write clearly in lab reports.** Explain what each result means in the model, not just how you got it.

Next, see the [MATH 212 guide](/courses/duke-math-212) for multivariable calculus.

## Frequently asked questions

### What is Duke MATH 112L?

MATH 112L, Laboratory Calculus II, is the second semester of introductory calculus with a laboratory component. It covers methods and applications of integration, improper integrals, probability, infinite series, Taylor polynomials and several kinds of differential equations.

### What are the prerequisites for MATH 112L?

MATH 106L or 111L, or consent. The department notes that AP or placement credit for MATH 21 is not sufficient if MATH 122L is offered.

### What is the difference between MATH 112L and MATH 122L?

MATH 112L continues MATH 106L or 111L. MATH 122L, Introductory Calculus II with Applications, is for students with advanced placement credit for MATH 21. A student cannot take both.

### Does MATH 112L cover Fourier series?

Yes. The department's description lists Fourier series along with series solutions and systems of differential equations.

### What comes after MATH 112L?

MATH 212, Multivariable Calculus, lists MATH 22, 112L, 122 or 122L as its prerequisite.