# CMU 21-122 Integration and Approximation: Topics and Practice

> A study guide for Carnegie Mellon 21-122 Integration and Approximation: integration techniques, series, Taylor series, ODEs and polar curves, with practice.

Source: https://practicecalculus.com/courses/carnegie-mellon-21-122
Updated: 2026-11-08

## At a glance

- **Course code:** 21-122
- **Units:** 10 (per the department's sample schedule)
- **Offered:** Spring of Year 1 in the department's sample schedule; other terms are not stated on the department pages
- **Prerequisites:** Not stated on the department topic list; it follows 21-120 in the department's sequence
- **Covers:** Integration techniques, improper integrals, numerical integration, sequences and series, power and Taylor series, Newton's method, arc length and surface area, separable differential equations, parametric curves and polar coordinates
- **Next course:** 21-259 (Calculus in Three Dimensions)
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- 21-122 is Carnegie Mellon's second calculus course, taken after 21-120.
- The official topic list covers integration techniques, improper integrals, numerical integration, series and Taylor series.
- It also includes Newton's method, arc length and surface area, separable differential equations, parametric curves and polar coordinates.
- Passing the 21-122 placement exam gives a prerequisite waiver, not course credit.

**CMU 21-122**, Integration and Approximation, is the second calculus course at Carnegie Mellon, following [21-120](/courses/carnegie-mellon-21-120). The Department of Mathematical Sciences publishes a full topic list for its placement exam, and it shows a course that combines integration techniques with a strong theme of approximation: numerical integration, series, Taylor polynomials and Newton's method. This guide maps every topic on that list to free practice.

## What 21-122 covers

The department's 21-122 topic list begins with integration: standard formulas, u-substitution, integration by parts, trig identities for products of powers of trig functions, trig substitution for $\sqrt{a^2 - x^2}$, $\sqrt{x^2 + a^2}$ and $\sqrt{x^2 - a^2}$, and partial fractions. Improper integrals follow, including the $p$-integral $\int_1^\infty \frac{1}{x^p}\,dx$ and the Comparison Theorem. The Midpoint and Trapezoid Rules come with their error bounds.

The longest part of the list is sequences and series: convergence of sequences with the Squeeze and Monotone Convergence Theorems, geometric series, the Test for Divergence, the Integral Test with remainder estimates, direct and limit comparison, the Alternating Series Test with its error bound, the Ratio and Root Tests, absolute and conditional convergence, power series with radius and interval of convergence, building series from $\frac{1}{1-x}$, differentiating and integrating series, Taylor and Maclaurin series, Taylor's Theorem with Remainder, and the binomial series.

The list ends with Newton's method, arc length and surfaces of revolution, basic and separable differential equations, parametric curves with their derivatives and arc length, and polar coordinates with area and arc length.

## 21-122 topic map

| 21-122 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 | [Trig integrals](/learn/calculus-2/integration-techniques/trig-integrals) | [Trig integrals practice](/practice-problems/trig-integrals) |
| Trig substitution | [Trig substitution](/learn/calculus-2/integration-techniques/trig-substitution) | [Trig substitution guide](/blog/trig-substitution) |
| Partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions guide](/blog/partial-fractions) |
| Improper integrals and comparison | [Comparison test](/learn/calculus-2/improper-integrals-and-applications/comparison-test) | [Improper integrals](/blog/improper-integrals) |
| Midpoint and Trapezoid Rules | [Trapezoid sums](/learn/calculus-1/integrals/trapezoid-sums) | [Riemann sums](/blog/riemann-sums) |
| Sequences | [Sequences](/learn/calculus-2/sequences-and-series/sequences) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Geometric series and divergence test | [Geometric series](/learn/calculus-2/sequences-and-series/geometric-series) | [nth term test](/rules/nth-term-test) |
| Integral and comparison tests | [Integral test and p-series](/learn/calculus-2/sequences-and-series/integral-test-and-p-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) |
| Ratio and root tests | [Ratio and root tests](/learn/calculus-2/sequences-and-series/ratio-and-root-tests) | [Ratio test](/rules/ratio-test) |
| Power series | [Power series and radius](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series) |
| Taylor series and remainder | [Estimating with Taylor](/learn/calculus-2/power-series-parametric-polar/estimating-with-taylor) | [Taylor's theorem](/theorems/taylors-theorem) |
| Arc length and surface area | [Arc length and surface area](/learn/calculus-2/improper-integrals-and-applications/arc-length-and-surface-area) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Separable differential equations | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Differential equations practice](/practice-problems/differential-equations) |
| Parametric curves | [Parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) | [Parametric and polar practice](/practice-problems/parametric-and-polar) |
| Polar coordinates | [Polar area](/learn/calculus-2/power-series-parametric-polar/polar-area) | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) |

Newton's method also appears on the list. It uses tangent lines, so the [tangent lines](/learn/calculus-1/the-derivative/tangent-lines) lesson is the best warm-up.

## Example problems at the 21-122 level

**1. Trig substitution.** With $x = \sin\theta$,

$$\int \frac{dx}{\sqrt{1 - x^2}} = \int d\theta = \arcsin x + C$$

**2. An alternating series bound.** For $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}$, the error after 9 terms is at most $\frac{1}{100}$.

**3. Newton's method.** To approximate $\sqrt{2}$, use $f(x) = x^2 - 2$ and $x_{n+1} = x_n - \frac{x_n^2 - 2}{2x_n}$. From $x_0 = 1$: $x_1 = 1.5$, $x_2 \approx 1.4167$.

## How to do well in 21-122

- **Practice choosing a method.** Use [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) until picking a technique is automatic.
- **Learn every error bound.** Trapezoid, alternating series, integral test and Taylor remainders all appear on the list.
- **Build series from known ones.** Starting from $\frac{1}{1-x}$ saves a lot of work on power series questions.
- **Keep 21-120 skills sharp.** L'Hospital's rule reappears in improper integrals, so revisit [21-120](/courses/carnegie-mellon-21-120) topics as needed.

## Frequently asked questions

### What is CMU 21-122?

21-122 Integration and Approximation is Carnegie Mellon's second-semester calculus course. Its topic list covers integration techniques, improper integrals, numerical integration, sequences and series, Taylor series, differential equations, and parametric and polar curves.

### How many units is 21-122?

The Department of Mathematical Sciences sample schedule lists 21-122 at 10 units.

### What does "approximation" mean in 21-122?

Approximation runs through the course: the Midpoint and Trapezoid Rules with error bounds, Taylor polynomials with remainder estimates, series remainder estimates, and Newton's method for roots.

### Can I place out of 21-122?

AP, IB or Cambridge scores can earn credit under CMU's credit rules. Passing the 21-122 placement exam during orientation gives a prerequisite waiver, not course credit.

### What comes after 21-122?

Many students continue to 21-259 Calculus in Three Dimensions, CMU's multivariable calculus course.