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Stanford MATH 20: Calculus

MATH 20 is the integral-calculus quarter of Stanford's single-variable sequence — the definite integral, the fundamental theorem, integration techniques, and applications — between MATH 19's derivatives and MATH 21's series. Many students enter via AP credit placement rather than MATH 19.

Fennie is independent and not affiliated with Stanford University. This is an unofficial study guide.

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What makes it hard

Integration is the first calculus skill that requires recognition, not just execution: knowing which technique an integral wants is learned only through mixed-problem volume. Students entering on AP credit sometimes discover their differentiation is rustier than their placement, and the ten-week clock leaves little room to rehab it mid-quarter.

What you'll cover

  • • The definite integral and Riemann sums
  • • The fundamental theorem of calculus
  • • Substitution and integration techniques
  • • Applications: area and volume
  • • Improper integrals

The MATH 20 study guide

How to study for Stanford MATH 20, step by step.

  1. 1

    Confirm your derivative fluency immediately

    Integration punishes rusty differentiation constantly — every substitution is a derivative read backward. If you placed in via AP credit, spend week one verifying the foundation is as solid as the placement assumed.

  2. 2

    Do mixed integral sets, not topic-sorted ones

    Recognizing which technique an integral wants is the exam skill, and topic-sorted homework can't build it. Add shuffled sets from week three onward.

  3. 3

    Understand the fundamental theorem, don't just use it

    Conceptual questions about what integration means and why the theorem connects it to derivatives are reliable exam material. Practice explaining it in plain words.

  4. 4

    Set up applications from scratch

    Area and volume problems fail at the setup — the sketch, the slice, the bounds. Practice the translation from picture to integral without a template in front of you.

  5. 5

    Rehearse timed before each exam

    Mixed integrals under time pressure, no notes. Technique selection at speed is what the exams measure, and untimed homework comfort doesn't transfer on its own.

  6. 6

    Build the recognition reps with Fennie

    Upload your MATH 20 syllabus and Fennie's Daily Plan schedules daily mixed-technique practice paced to the midterms, with derivative refreshers built in and quizzes from the actual course content. It's free to start.

    Start my MATH 20 plan free

How Fennie helps with MATH 20

Fennie's Daily Plans build the mixed-integral volume MATH 20 actually requires — technique recognition comes only from shuffled daily practice, which the plan schedules around the midterm dates. Chat through why an integral wants substitution versus parts until the choice is reflexive, with derivative rehab woven in for AP-credit arrivals.

FAQ

Is MATH 20 hard?

Moderately — integration demands pattern recognition that only practice volume builds, on a ten-week clock. The common stumble is students entering via AP credit with rustier derivatives than their placement suggested. Daily mixed practice handles both problems.

Can I skip MATH 20 with AP credit?

Strong BC scores typically place past it; AB credit usually lands you here. Follow the math department's placement guidance honestly — placing forward with a shaky integration foundation makes MATH 21 and MATH 51 harder than they need to be.

How do I study for MATH 20 exams?

Mixed integral sets under time limits — the exams test choosing the technique as much as executing it. Practice application setups (areas, volumes) from the picture up, and keep differentiation warm since every substitution leans on it.

Pass MATH 20 with a plan, not a cram

Upload your MATH 20 materials and Fennie generates a Daily Plan paced to your deadline — plus chat, flashcards, and quizzes built from the actual course content.

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More Stanford courses

MATH 51 — Linear Algebra, Multivariable Calculus, and Modern Applications

MATH 51 is Stanford's famous hybrid — linear algebra and multivariable differential calculus taught as one integrated course from an in-house textbook — required for CS, engineering, and most quantitative majors. It's many students' first encounter with college math at Stanford pace.

MATH 19 — Calculus

MATH 19 opens Stanford's single-variable calculus sequence — limits, continuity, and differential calculus with a careful treatment of the functions underneath — for students starting calculus at Stanford rather than placing past it. It runs on the same ten-week clock as everything else.

MATH 21 — Calculus

MATH 21 completes Stanford's single-variable sequence with sequences and series — convergence tests, power series, and Taylor series — the standard final gate before MATH 51. It's widely considered the most conceptually demanding quarter of the three.