# UCSD MATH 20B Calculus: Topics, Study Guide and Practice

> A study guide for UCSD MATH 20B: integration methods, infinite series, polar coordinates and complex exponentials, mapped to free practice.

Source: https://practicecalculus.com/courses/uc-san-diego-math-20b
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 20B
- **Units:** 4
- **Prerequisites:** AP Calculus AB 4 or 5, AP Calculus BC 3, or a C- or better in MATH 20A, 10B or 10C
- **Covers:** Integral calculus of one variable with exponential, log, hyperbolic and trig functions, methods of integration, infinite series, polar coordinates and complex exponentials
- **Credit note:** Two units if taken after MATH 10B or 10C
- **Next course:** MATH 20C (Calculus and Analytic Geometry for Science and Engineering)
- **Practice here:** [Calculus 1](/learn/calculus-1) integrals units plus [Calculus 2](/learn/calculus-2)

## Key takeaways

- MATH 20B is the second quarter of UC San Diego's science and engineering calculus sequence, worth 4 units.
- It is an integral calculus course: definite integrals and their applications, integration methods, infinite series and polar coordinates.
- It also introduces complex exponentials, a topic many Calculus 2 courses skip.
- A C- or better in MATH 20A, a 4 or 5 on AP Calculus AB, or a 3 on AP Calculus BC lets you enroll.

**MATH 20B** is the second quarter of UC San Diego's calculus sequence for science and engineering. It picks up where MATH 20A stops, at the integral, and carries students through integration methods, infinite series and polar coordinates. This guide maps every MATH 20B topic to free practice so you can learn each technique by working problems with instant feedback.

## What UCSD MATH 20B covers

According to the UC San Diego catalog, MATH 20B is integral calculus of one variable and its applications. It works with exponential, logarithmic, hyperbolic and trigonometric functions, develops methods of integration, and then turns to infinite series, polar coordinates in the plane and complex exponentials.

At most universities this material is split between the end of **Calculus 1** (the definite integral and the Fundamental Theorem) and **Calculus 2** (techniques, series and polar coordinates). MATH 20B covers both pieces in one quarter, and adds complex exponentials such as $e^{i\theta} = \cos\theta + i\sin\theta$.

## MATH 20B topic map

Each row links to a lesson you can practice and to a reference page or guide that explains the idea.

| MATH 20B topic | Practice | Read |
| --- | --- | --- |
| The definite integral | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| Fundamental Theorem of Calculus | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus), [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1) |
| Substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [U-substitution](/blog/u-substitution), [substitution rule](/rules/u-substitution-rule) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts), [the formula](/rules/integration-by-parts-rule) |
| 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) |
| Hyperbolic functions | [Choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) | [Hyperbolic functions](/functions/hyperbolic-functions) |
| Applications: area and volume | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves](/blog/area-between-curves), [volumes of revolution](/blog/volumes-of-revolution) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Infinite series | [Geometric series](/learn/calculus-2/sequences-and-series/geometric-series) | [Geometric series](/blog/geometric-series), [convergence tests](/blog/convergence-tests) |
| Power and Taylor series | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series), [power series](/blog/power-series) |
| Polar coordinates | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) | [Area in polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-area) |

## MATH 20B vs MATH 20A and MATH 20C

| Course | Main idea | What you should leave with |
| --- | --- | --- |
| [MATH 20A](/courses/uc-san-diego-math-20a) | Derivatives | Limits, derivative rules, optimization |
| MATH 20B | Integrals and series | Integration methods, convergence, polar curves |
| [MATH 20C](/courses/uc-san-diego-math-20c) | Several variables | Vectors, partial derivatives, double integrals |

MATH 20B is where integration becomes a toolbox rather than a single rule. Series also matter later: Taylor expansions return in differential equations and physics. When you finish, the [MATH 20C guide](/courses/uc-san-diego-math-20c) covers the move to several variables.

## Example problems at the MATH 20B level

**1. Integration by parts.** Evaluate $\int x\cos x\,dx$ with $u = x$ and $dv = \cos x\,dx$, so $du = dx$ and $v = \sin x$:

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

**2. A geometric series.** Find the sum of $\sum_{n=0}^{\infty} \frac{3}{4^n}$.

This is geometric with first term $a = 3$ and ratio $r = \frac{1}{4}$, and $|r| < 1$, so

$$\sum_{n=0}^{\infty} \frac{3}{4^n} = \frac{3}{1 - \frac{1}{4}} = 4$$

**3. A complex exponential.** Write $e^{i\pi/2}$ in the form $a + bi$.

By Euler's formula, $e^{i\pi/2} = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} = 0 + i = i$.

## How to do well in UCSD MATH 20B

- **Practice choosing the method.** Exams rarely tell you which technique to use. The [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) lesson and mixed [practice exams](/practice) train that decision.
- **Memorize the core integrals.** The [calculus formula sheet](/formulas) lists every basic integral and series you need.
- **Learn the geometric series cold.** Most series arguments compare to it, so start with the [geometric series formula](/rules/geometric-series-formula).
- **Check answers by differentiating.** Differentiate your antiderivative to catch sign errors, one of the [common calculus mistakes](/blog/common-calculus-mistakes) that costs the most points.

> **Tip:** for series, ask the same three questions every time: does the term go to zero, does it look like a geometric or $p$-series, and does the ratio test apply?

## Frequently asked questions

### What is MATH 20B at UCSD?

MATH 20B is UC San Diego's second-quarter calculus course for science and engineering. It covers integral calculus of one variable and its applications, methods of integration, infinite series, polar coordinates in the plane and complex exponentials.

### What comes after MATH 20B?

MATH 20C, Calculus and Analytic Geometry for Science and Engineering, covers vectors, vector functions, partial derivatives, maxima and minima and double integrals.

### Is MATH 20B harder than MATH 20A?

Many students find it harder, because integration requires choosing a technique rather than following one rule, and series introduce new ways of thinking about convergence. Regular mixed practice, where you decide which method to use, helps more than anything else.

### Can I skip MATH 20A with AP credit?

The catalog lists a 4 or 5 on AP Calculus AB, or a 3 on AP Calculus BC, as meeting the MATH 20B prerequisite. Check with UCSD advising for how your scores apply to your major.

### How should I prepare for MATH 20B?

Make sure derivatives are automatic and review antiderivatives, Riemann sums and u-substitution. Then start integration by parts and the geometric series, which come up early.