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.
Stanford University Updated 3 min read
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 integrals units and the start of Calculus 2
On this page
Stanford Math 20 is the middle course of Stanford's three-quarter single-variable calculus series. After 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 | Riemann sums guide |
| The definite integral | The definite integral as area | Definite integral properties |
| Antiderivatives | Undoing derivatives | Antiderivatives guide, reverse power rule |
| Fundamental Theorem of Calculus | Evaluating definite integrals | FTC guide, FTC part 1 |
| Substitution | U-substitution | U-substitution guide |
| Integration by parts | Integration by parts | Integration by parts guide |
| Area between curves | Area between curves | Area between curves guide |
| Volumes by slices and washers | Volumes with washers | Volumes of revolution |
| Initial-value problems and growth models | Separable differential equations | Exponential functions |
| Parametric curves and arc length | Parametric curves | Arc length of parametric curves |
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 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:
2. Volume by shells. Rotate the region under y = x^2 for 0 \le x \le 1 about the y-axis:
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 set until it is quick.
- Draw the region first for every area and volume problem, then decide between washers and shells. Our volumes guide 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 and timed practice exams, and read the common calculus mistakes before each midterm.
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Area with rectangles
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