# Stanford Math 20 Calculus: Topics, Study Guide and Practice

> A study guide for Stanford Math 20: Riemann sums, the Fundamental Theorem, substitution, parts, volumes and differential equations, with free practice.

Source: https://practicecalculus.com/courses/stanford-math-20
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 20
- **Units:** 3
- **Offered:** Autumn, winter and spring quarters (2026-27)
- **Covers:** The definite integral, Riemann sums, antiderivatives, the Fundamental Theorem, substitution and integration by parts, areas, volumes by slices, washers and shells, initial-value problems, exponential and logistic models, direction fields and parametric curves
- **Previous course:** MATH 19 (Calculus)
- **Next course:** MATH 21 (Calculus)
- **Practice here:** [Calculus 1](/learn/calculus-1) integrals units and the start of [Calculus 2](/learn/calculus-2)

## Key takeaways

- Math 20 is the integration course in Stanford's single-variable series, between Math 19 and Math 21.
- It covers the definite integral and Fundamental Theorem, substitution and integration by parts, and areas and volumes.
- It also introduces differential equations through initial-value problems, exponential and logistic models, and direction fields.
- It is a 3-unit course offered in all three regular quarters.

**Stanford Math 20** is the middle course of Stanford's three-quarter single-variable calculus series. After [Math 19](/courses/stanford-math-19) builds the derivative, Math 20 builds the integral and uses it to measure areas and volumes and to model change with differential equations. This guide maps each Math 20 topic to free practice with instant feedback.

## What Stanford Math 20 covers

Stanford's course catalog lists the definite integral, Riemann sums, antiderivatives and the Fundamental Theorem of Calculus, followed by integration by substitution and by parts. Applications include the area between curves and volumes computed with slices, washers and shells. The course closes with initial-value problems, exponential and logistic models, direction fields, and parametric curves. The math department's overview also mentions arc length of parametric curves and separable equations.

Against a semester-based sequence, Math 20 combines the integration end of **Calculus 1** with the first parts of **Calculus 2**: integration by parts, shell volumes, and simple differential equations.

## Math 20 topic map

| Math 20 topic | Practice | Read |
| --- | --- | --- |
| Riemann sums | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums guide](/blog/riemann-sums) |
| The definite integral | [The definite integral as area](/learn/calculus-1/integrals/signed-area) | [Definite integral properties](/rules/definite-integral-properties) |
| Antiderivatives | [Undoing derivatives](/learn/calculus-1/integrals/undoing-derivatives) | [Antiderivatives guide](/blog/antiderivatives), [reverse power rule](/rules/reverse-power-rule) |
| Fundamental Theorem of Calculus | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [FTC guide](/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 guide](/blog/u-substitution) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts guide](/blog/integration-by-parts) |
| Area between curves | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves guide](/blog/area-between-curves) |
| Volumes by slices and washers | [Volumes with washers](/learn/calculus-1/applications-of-integrals/volumes-washers) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Initial-value problems and growth models | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Exponential functions](/functions/exponential-functions) |
| Parametric curves and arc length | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) | [Arc length of parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) |

## Math 19 vs Math 20 vs Math 21

| Course | Units | Offered (2026-27) | Focus |
| --- | --- | --- | --- |
| Math 19 | 3 | Autumn, winter | Limits and derivatives |
| Math 20 | 3 | Autumn, winter, spring | Integrals and their applications, intro to differential equations |
| Math 21 | 4 | Autumn, winter, spring | Infinite processes: improper integrals, series, Taylor series |

Rational functions are a loose end that the catalog says [Math 21](/courses/stanford-math-21) picks up, so partial fractions arrive in the next course.

## Example problems at the Math 20 level

**1. Substitution.** With $u = x^2 + 1$, $du = 2x\,dx$:

$$\int_0^1 2x(x^2 + 1)^3\,dx = \int_1^2 u^3\,du = \frac{u^4}{4}\Big|_1^2 = 4 - \frac{1}{4} = \frac{15}{4}$$

**2. Volume by shells.** Rotate the region under $y = x^2$ for $0 \le x \le 1$ about the $y$-axis:

$$V = \int_0^1 2\pi x \cdot x^2\,dx = 2\pi \cdot \frac{1}{4} = \frac{\pi}{2}$$

**3. An initial-value problem.** Solve $y' = -0.5y$, $y(0) = 8$. The solution is $y = 8e^{-0.5t}$. The half-life solves $e^{-0.5t} = \frac{1}{2}$, so $t = 2\ln 2 \approx 1.39$, when $y = 4$.

## How to do well in Stanford Math 20

- **Make antiderivatives automatic.** Every technique ends with a basic antiderivative. Drill the [antiderivative practice](/learn/calculus-1/integrals/antiderivative-practice) set until it is quick.
- **Draw the region first** for every area and volume problem, then decide between washers and shells. Our [volumes guide](/blog/volumes-of-revolution) compares them.
- **Check answers by differentiating.** Any indefinite integral can be verified with the derivative rules from Math 19, which catches most sign errors.
- **Use the [formula sheet](/formulas)** and timed [practice exams](/practice), and read the [common calculus mistakes](/blog/common-calculus-mistakes) before each midterm.

## Frequently asked questions

### What is Stanford Math 20?

Math 20 is the second quarter of single-variable calculus at Stanford. It introduces the definite integral through Riemann sums, connects it to antiderivatives with the Fundamental Theorem of Calculus, teaches substitution and integration by parts, and applies integrals to areas and volumes. It ends with differential equation models and parametric curves.

### What comes after Math 20?

Math 21, which covers limits at infinity, sequences, improper integrals, infinite series and Taylor series.

### Can I take Math 20 without Math 19?

The math department says students with sufficient background can start the 20-series at any point. If you already know derivatives well, for example from a high school course, check the department placement guidance.

### Is Math 20 harder than Math 19?

Many students find integration harder than differentiation because there is no single rule that always works. Practicing many short substitution and parts problems is the fastest way to build that judgment.