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.
NYU Updated 3 min read
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
On this page
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 | Integration by parts guide |
| Trig integrals and substitution | Trig substitution | Trig substitution |
| Partial fractions | Partial fractions | Partial fractions |
| Improper integrals | Integrating to infinity | Improper integrals |
| Further applications | Arc length and surface area | Volumes of revolution |
| Sequences and series | Series and partial sums | Geometric series, nth-term test |
| Convergence tests | Choosing a test | Convergence tests, ratio test |
| Power and Taylor series | Power series and radius | Power series, Taylor series |
| Parametric equations | Parametric curves | Implicit differentiation |
| Polar coordinates | Area in polar coordinates | Area between curves |
MATH-UA 122 in the NYU calculus sequence
| Course | Focus | How students usually enter |
|---|---|---|
| MATH-UA 121 Calculus I | 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 | 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:
2. A geometric series.
3. A radius of convergence. For \sum_{n=1}^{\infty} \frac{x^n}{n\,3^n}, the ratio test gives
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 train the decision that exams actually test.
- Anchor series on two families. Know the 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 lesson drills them.
- Keep the formula sheet open for trig identities while you learn trig integrals, and take timed practice exams before each midterm.
- Review the classic errors in 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.
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Integration by parts
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