# Columbia MATH UN1102 Calculus II: Topics, Study Guide and Practice

> A study guide for Columbia MATH UN1102 Calculus II: integration methods, applications of the integral, Taylor's theorem and series, with practice.

Source: https://practicecalculus.com/courses/columbia-math-un1102
Updated: 2026-11-03

## At a glance

- **Course code:** MATH UN1102
- **Points:** 3
- **Offered:** Fall and spring
- **Prerequisites:** MATH UN1101 or the equivalent
- **Covers:** Methods of integration, applications of the integral, Taylor's theorem, infinite series
- **Next course:** MATH UN1201 Calculus III, UN1205 Accelerated Multivariable Calculus, or UN1207 Honors Mathematics A
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- MATH UN1102 is Columbia's Calculus II, a 3-point course offered in fall and spring with UN1101 or equivalent as the prerequisite.
- It covers methods of integration, applications of the integral, Taylor's theorem and infinite series.
- It is required before UN1202 Calculus IV and is a prerequisite for UN1205.
- The department strongly urges taking it before Calculus III, even though it is no longer a formal prerequisite.

**Columbia MATH UN1102** is Calculus II at Columbia University. It finishes single-variable calculus and appears in every one of the department's calculus sequences, before multivariable calculus in UN1201, UN1205 or the honors track. This guide matches each UN1102 topic with a lesson you can practice with instant feedback.

## What Columbia MATH UN1102 covers

The Columbia bulletin lists methods of integration, applications of the integral, Taylor's theorem and infinite series. The prerequisite is MATH UN1101 or the equivalent.

That is the core of a standard **Calculus 2**: integration techniques, applications such as volume and arc length, convergence of series, and Taylor polynomials and series. Polar and parametric curves are not named in the bulletin description. If your section covers them, the power series, parametric and polar unit on PracticeCalculus has lessons for both.

The two halves of the course feel quite different. The integration half rewards pattern recognition: seeing which method fits an integrand. The series half rewards careful reasoning about limits: deciding whether an infinite sum settles down, and how well a Taylor polynomial approximates a function. Practicing both kinds of problem every week, rather than one unit at a time, makes the final exam much less of a shock.

## UN1102 topic map

| UN1102 topic | Practice | Read |
| --- | --- | --- |
| Substitution review | [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) |
| Trig integrals and substitution | [Trig substitution](/learn/calculus-2/integration-techniques/trig-substitution) | [Trig substitution](/blog/trig-substitution) |
| Partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions](/blog/partial-fractions) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Applications of the integral | [Work](/learn/calculus-2/improper-integrals-and-applications/work) | [Volumes of revolution](/blog/volumes-of-revolution), [area between curves](/blog/area-between-curves) |
| Infinite series | [Geometric series](/learn/calculus-2/sequences-and-series/geometric-series) | [Geometric series](/blog/geometric-series), [convergence tests](/blog/convergence-tests) |
| Alternating series | [Alternating series](/learn/calculus-2/sequences-and-series/alternating-series) | [Alternating series test](/rules/alternating-series-test) |
| Taylor polynomials | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series) |
| Taylor's theorem | [Estimating with Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/estimating-with-taylor) | [Taylor's theorem](/theorems/taylors-theorem) |

## Where UN1102 leads

| Next course | What it covers | Notes from the bulletin |
| --- | --- | --- |
| [UN1201 Calculus III](/courses/columbia-math-un1201) | Vectors, partial derivatives, optimization, Lagrange multipliers | UN1102 strongly urged first |
| UN1205 Accelerated Multivariable Calculus | UN1201 and UN1202 in one course | Requires UN1101 and UN1102 |
| UN1207 and UN1208 Honors Mathematics A and B | Rigorous multivariable calculus and linear algebra | Recommended for math majors |

Calculus II is also required before [UN1202 Calculus IV](/courses/columbia-math-un1202). The course before it is [UN1101 Calculus I](/courses/columbia-math-un1101).

## Example problems at the UN1102 level

**1. Trig substitution.** With $x = 2\sin\theta$, $dx = 2\cos\theta\,d\theta$ and $\sqrt{4 - x^2} = 2\cos\theta$:

$$\int \frac{dx}{\sqrt{4 - x^2}} = \int d\theta = \theta + C = \arcsin\frac{x}{2} + C$$

**2. Estimating with a Taylor polynomial.** The degree-3 Maclaurin polynomial of $\ln(1 + x)$ is $x - \frac{x^2}{2} + \frac{x^3}{3}$, so

$$\ln(1.1) \approx 0.1 - 0.005 + 0.000333 \approx 0.09533$$

which matches the true value $0.09531$ to four decimal places.

**3. An alternating series.** $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges by the alternating series test, because $\frac{1}{n}$ decreases to $0$. It converges conditionally, since $\sum \frac{1}{n}$ diverges.

## How to do well in MATH UN1102

- **Practice picking the method.** The [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) lesson mixes techniques the way exams do.
- **Learn series tests in a fixed order:** nth-term, geometric or p-series, comparison, ratio, alternating. The [choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) lesson drills that order.
- **Memorize the core Maclaurin series** for $e^x$, $\sin x$, $\cos x$, $\ln(1 + x)$ and $\frac{1}{1-x}$. Most Taylor problems are built from them.
- **Use the [formula sheet](/formulas)** for trig identities, scan [common calculus mistakes](/blog/common-calculus-mistakes), and take timed [practice exams](/practice).

> **Tip:** the error in an alternating series estimate is at most the first term you left out. It is the fastest error bound in the course.

## Frequently asked questions

### What is Columbia MATH UN1102?

MATH UN1102 is Calculus II at Columbia. It covers methods of integration, applications of the integral, Taylor's theorem and infinite series, and is worth 3 points.

### Do I need UN1102 before Calculus III?

The bulletin says Calculus II is no longer a prerequisite for UN1201 Calculus III, but the department strongly urges students to take it first, especially prospective math majors. It is required before UN1202 Calculus IV and for UN1205.

### What comes after UN1102?

Most students take UN1201 Calculus III and then UN1202 Calculus IV. Alternatives are UN1205 Accelerated Multivariable Calculus, or the honors pair UN1207 and UN1208.

### Is Calculus II harder than Calculus I?

Many students find it the toughest calculus course, because both integration methods and series tests require choosing an approach rather than applying one rule. Regular mixed practice is the most reliable fix.

### How do I prepare for UN1102?

Make sure antiderivatives, u-substitution and the Fundamental Theorem from UN1101 are automatic. Then preview integration by parts, which is usually the first new method.