# UC Davis MAT 21B Calculus: Topics, Study Guide and Practice

> A study guide for UC Davis MAT 21B: definite integrals, the Fundamental Theorem, integration techniques, volume, arc length and improper integrals.

Source: https://practicecalculus.com/courses/uc-davis-mat-21b
Updated: 2026-11-03

## At a glance

- **Course code:** MAT 021B (honors version: MAT 021BH)
- **Units:** 4
- **Prerequisites:** MAT 021A or 021AH with a C- or better, or MAT 017A or 019A with a B or better
- **Covers:** Definite integrals, the Fundamental Theorem of Calculus, integration techniques, area, volume, arc length, average value, improper integrals and surfaces of revolution
- **Credit note:** Only 2 units if MAT 016B, 016C, 017B or 017C was completed
- **Next course:** MAT 021C (Calculus)
- **Practice here:** [Calculus 1](/learn/calculus-1) integrals units plus [Calculus 2](/learn/calculus-2) integration units

## Key takeaways

- MAT 21B is the second quarter of the UC Davis calculus sequence and is devoted to integration.
- It starts with the definite integral and the Fundamental Theorem, then covers integration techniques and their applications.
- Applications include area, volume, arc length, average value, surfaces of revolution and improper integrals.
- Series are not part of 21B; they open MAT 21C.

**MAT 21B** is the second quarter of the UC Davis calculus sequence, and it is all about integration. Students arrive from MAT 21A knowing derivatives and leave able to compute areas, volumes and arc lengths, and to handle integrals over infinite intervals. This guide maps every MAT 21B topic to free practice so you can work each technique until it sticks.

## What UC Davis MAT 21B covers

The UC Davis General Catalog lists definite integrals, the Fundamental Theorem of Calculus and techniques of integration as the core of MAT 21B. The rest of the course applies those tools: area, volume, arc length, average value, improper integrals and surfaces of revolution.

In semester terms, MAT 21B combines the integration chapters at the end of **Calculus 1** with the integration techniques and applications at the start of **Calculus 2**. Infinite series, which many semester schools also put in Calculus 2, are covered in MAT 21C instead.

## MAT 21B topic map

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

| MAT 21B topic | Practice | Read |
| --- | --- | --- |
| Antiderivatives | [Undoing derivatives](/learn/calculus-1/integrals/undoing-derivatives) | [Antiderivatives](/blog/antiderivatives), [reverse power rule](/rules/reverse-power-rule) |
| Riemann sums and the definite integral | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums), [definite integral properties](/rules/definite-integral-properties) |
| 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 2](/theorems/fundamental-theorem-of-calculus-part-2) |
| Substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [U-substitution](/blog/u-substitution) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts) |
| Trig integrals and trig substitution | [Powers of sine and cosine](/learn/calculus-2/integration-techniques/trig-integrals) | [Trig substitution](/blog/trig-substitution) |
| Partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions](/blog/partial-fractions) |
| Area | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves](/blog/area-between-curves) |
| Volume | [Volumes with washers](/learn/calculus-1/applications-of-integrals/volumes-washers) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Average value | [Average value](/learn/calculus-1/applications-of-integrals/average-value) | [Mean Value Theorem for integrals](/theorems/mean-value-theorem-for-integrals) |
| Arc length and surfaces of revolution | [Arc length and surface area](/learn/calculus-2/improper-integrals-and-applications/arc-length-and-surface-area) | [Definite integral properties](/rules/definite-integral-properties) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |

## MAT 21B in the UC Davis sequence

| Course | Focus | Honors version |
| --- | --- | --- |
| [MAT 21A](/courses/uc-davis-mat-21a) | Limits and derivatives | MAT 21AH |
| MAT 21B | Integration and its applications | MAT 21BH |
| [MAT 21C](/courses/uc-davis-mat-21c) | Series, vectors, partial derivatives | MAT 21CH |
| [MAT 21D](/courses/uc-davis-mat-21d) | Vector analysis | None listed |

Students coming from the life-science track can enter 21B with a B or better in MAT 17A. The honors version, 21BH, requires a B or better in 21A or 21AH. After this course, the [MAT 21C guide](/courses/uc-davis-mat-21c) covers series and the start of multivariable calculus.

## Example problems at the MAT 21B level

**1. The Fundamental Theorem.** Evaluate $\int_0^{\pi} \sin x\,dx$.

$$\int_0^{\pi} \sin x\,dx = \left[-\cos x\right]_0^{\pi} = -(-1) - (-1) = 2$$

**2. A volume with disks.** Rotate $y = \sqrt{x}$ for $0 \le x \le 4$ about the $x$-axis. Each disk has radius $\sqrt{x}$, so

$$V = \pi \int_0^4 (\sqrt{x})^2\,dx = \pi \int_0^4 x\,dx = \pi \cdot 8 = 8\pi$$

**3. An improper integral.** Evaluate $\int_0^{\infty} e^{-2x}\,dx$.

$$\lim_{t \to \infty} \left[-\frac{1}{2}e^{-2x}\right]_0^t = \lim_{t \to \infty} \left(\frac{1}{2} - \frac{1}{2}e^{-2t}\right) = \frac{1}{2}$$

## How to do well in UC Davis MAT 21B

- **Practice choosing a technique.** Exams mix substitution, parts and partial fractions without labels. The [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) lesson and mixed [practice exams](/practice) build that skill.
- **Draw every application.** For area and volume, sketch the region and a typical slice before writing an integral. It prevents most setup mistakes.
- **Check by differentiating.** Differentiate each antiderivative you find; it catches the sign and constant errors listed in [common calculus mistakes](/blog/common-calculus-mistakes).
- **Memorize the core integrals.** The [calculus formula sheet](/formulas) puts them all on one page.

> **Tip:** with integration by parts, choose $u$ as the factor that gets simpler when differentiated. The [integration by parts rule](/rules/integration-by-parts-rule) page shows the LIATE order.

## Frequently asked questions

### What is MAT 21B at UC Davis?

MAT 21B is the second course in UC Davis's main calculus sequence. It covers definite integrals, the Fundamental Theorem of Calculus and techniques of integration, with applications to area, volume, arc length, average value, improper integrals and surfaces of revolution.

### What do I need to take MAT 21B?

The catalog lists MAT 21A or 21AH with a C- or better, or MAT 17A or 19A with a B or better.

### Is MAT 21B harder than MAT 21A?

Many students find it harder, because integration requires choosing a method rather than applying a fixed rule, and applications like volumes require setting up the integral yourself. Mixed practice where you choose the technique is the best remedy.

### What comes after MAT 21B?

MAT 21C, which covers sequences and series, Taylor expansions, vectors, partial derivatives and optimization in several variables.

### How should I prepare for MAT 21B?

Make derivatives automatic, because every antiderivative is a derivative read backwards. Then practice basic antiderivatives and u-substitution before the quarter starts.