# NYU MATH-UA 122 Calculus II: Topics, Study Guide and Practice

> A study guide for NYU MATH-UA 122 Calculus II: integration techniques, polar and parametric equations and power series, with practice for each topic.

Source: https://practicecalculus.com/courses/nyu-math-ua-122
Updated: 2026-11-03

## At a glance

- **Course code:** MATH-UA 122
- **Credits:** 4
- **Offered:** Fall and spring
- **Prerequisites:** A grade of C or better in MATH-UA 121, a 4 or higher on AP Calculus AB or BC, or a qualifying placement score
- **Covers:** Techniques of integration, further applications of the integral, plane analytic geometry, polar and parametric equations, infinite series and power series
- **Next course:** MATH-UA 123 Calculus III
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- MATH-UA 122 is NYU's Calculus II, a 4-credit course offered in fall and spring.
- It covers integration techniques, more applications of integrals, polar and parametric equations, and infinite series including power series.
- You can enter with a C in MATH-UA 121, a 4 or 5 on AP Calculus AB or BC, or a placement score.
- It leads to MATH-UA 123 Calculus III, the multivariable course.

**NYU MATH-UA 122** is Calculus II at NYU, the second course in the College of Arts and Science calculus sequence. Students arrive from MATH-UA 121 or with AP credit, and leave with the integration and series skills needed for multivariable calculus, physics and engineering. Below, every MATH-UA 122 topic is matched to a lesson you can practice with instant feedback.

## What NYU MATH-UA 122 covers

According to the NYU bulletin, MATH-UA 122 covers techniques of integration and further applications of the integral, plane analytic geometry, polar and parametric equations, and infinite series, including power series.

That lines up with a standard **Calculus 2** course: the integration techniques unit, applications such as arc length and volume, then sequences and series, and finally power series together with parametric and polar curves. The Calculus 2 path on PracticeCalculus follows the same order.

## MATH-UA 122 topic map

| MATH-UA 122 topic | Practice | Read |
| --- | --- | --- |
| 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) |
| Further applications | [Arc length and surface area](/learn/calculus-2/improper-integrals-and-applications/arc-length-and-surface-area) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Sequences and series | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) | [Geometric series](/blog/geometric-series), [nth-term test](/rules/nth-term-test) |
| Convergence tests | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests), [ratio test](/rules/ratio-test) |
| Power and Taylor series | [Power series and radius](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series), [Taylor series](/blog/taylor-series) |
| Parametric equations | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) | [Implicit differentiation](/rules/implicit-differentiation) |
| Polar coordinates | [Area in polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-area) | [Area between curves](/blog/area-between-curves) |

## MATH-UA 122 in the NYU calculus sequence

| Course | Focus | How students usually enter |
| --- | --- | --- |
| [MATH-UA 121 Calculus I](/courses/nyu-math-ua-121) | Derivatives, integrals, areas and volumes | MATH-UA 9 or placement |
| MATH-UA 122 Calculus II | Techniques, polar and parametric, series | MATH-UA 121, AP AB/BC 4 or 5, or placement |
| [MATH-UA 123 Calculus III](/courses/nyu-math-ua-123) | Multivariable and vector calculus | MATH-UA 122, AP BC 5, or placement |

If you earned a 4 on the AP exam and are starting in MATH-UA 122, review u-substitution and the Fundamental Theorem first. The course assumes them from day one.

## Example problems at the MATH-UA 122 level

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

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

**2. A geometric series.**

$$\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n = \frac{1}{1 - \frac{2}{3}} = 3$$

**3. A radius of convergence.** For $\sum_{n=1}^{\infty} \frac{x^n}{n\,3^n}$, the ratio test gives

$$\lim_{n \to \infty} \left|\frac{x^{n+1}}{(n+1)3^{n+1}} \cdot \frac{n\,3^n}{x^n}\right| = \frac{|x|}{3}$$

The series converges when $|x| < 3$, so the radius of convergence is $3$.

## How to do well in MATH-UA 122

- **Practice choosing, not just computing.** Mixed sets like [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) train the decision that exams actually test.
- **Anchor series on two families.** Know the [geometric series formula](/rules/geometric-series-formula) and p-series cold. Nearly every comparison argument uses one of them.
- **Memorize a handful of Maclaurin series** for $e^x$, $\sin x$, $\cos x$ and $\frac{1}{1-x}$. The [common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) lesson drills them.
- **Keep the [formula sheet](/formulas) open** for trig identities while you learn trig integrals, and take timed [practice exams](/practice) before each midterm.
- **Review the classic errors** in [common calculus mistakes](/blog/common-calculus-mistakes), especially sign errors in integration by parts.

> **Tip:** for any series, try the nth-term test first. If the terms do not go to zero, you are done in one line.

## Frequently asked questions

### What is NYU MATH-UA 122?

MATH-UA 122 is Calculus II at NYU. It continues single-variable calculus with techniques of integration, further applications, plane analytic geometry, polar and parametric equations, and infinite series, including power series.

### Can I skip MATH-UA 121 and start in MATH-UA 122?

The NYU bulletin lets students start in MATH-UA 122 with a score of 4 or 5 on AP Calculus AB or BC, or with a qualifying placement score. Otherwise you need a C or better in MATH-UA 121.

### What comes after MATH-UA 122?

MATH-UA 123 Calculus III, which covers functions of several variables, vectors, partial derivatives, multiple integrals and the theorems of Gauss and Stokes. Students with a strong grade can consider the honors version, MATH-UA 129, when it is offered.

### Is Calculus II at NYU harder than Calculus I?

Many students find Calculus II the hardest course in a calculus sequence, because integration techniques and series tests require choosing a method rather than following one rule. Lots of mixed practice, where you have to decide which technique fits, is the best preparation.

### How do I get better at series in MATH-UA 122?

Learn the geometric series and p-series first, since most comparison arguments rely on them, then practice picking a convergence test from a mixed set of problems. Our convergence tests guide has a decision chart.