CMU 21-122: Integration and Approximation
21-122 is CMU's second calculus course: integration techniques, applications, improper integrals, then sequences, series, and Taylor approximation. It's where most AP-credit students enter the math sequence, and it's widely considered the harder half of first-year calculus.
Fennie is independent and not affiliated with Carnegie Mellon University. This is an unofficial study guide.
What makes it hard
Two distinct walls: integration technique selection, which only mixed-problem volume builds, and the series unit, which is more logic than computation and unlike anything before it. AP-credit entrants add a third hazard: discovering their high-school integration was thinner than their placement suggested, at CMU exam pace.
What you'll cover
- • Techniques of integration
- • Applications of integration
- • Improper integrals
- • Sequences and series
- • Convergence tests
- • Taylor and power series
The 21-122 study guide
How to study for CMU 21-122, step by step.
- 1
Stress-test your AP foundation immediately
If you placed in via AP credit, week one is for honestly verifying your integration and differentiation hold at CMU pace. Placement is a permission slip, not a guarantee.
- 2
Do mixed integral sets from the start
Knowing which technique an integral wants (parts, substitution, partial fractions) is the exam skill, and topic-sorted homework can't build it. Shuffle your practice deliberately.
- 3
Give series double the runway
Convergence reasoning is a conceptual leap that needs repeated exposures, not one strong week. Start reading ahead before the unit opens and revisit it often.
- 4
Build a convergence-test decision chart
Each test, its hypotheses, the series shapes it handles: one page. Practice classifying series rapidly with it, then without it. Exams grade the choice and the justification together.
- 5
Rehearse timed mixed exams
Before each exam, timed sets mixing techniques, applications, and series questions. Selection speed under pressure is what's measured, and untimed comfort doesn't transfer.
Today
Today's 21-122 plan
What a Fennie Daily Plan looks like for 21-122. Yours is built from your own syllabus and adapts every day to your deadlines and progress.
First plan free, no card required. Fennie is independent and unaffiliated with your school.
FAQ
Is 21-122 harder than 21-120?
Most students say yes: integration demands pattern recognition only volume builds, and series is a conceptual leap toward logic over computation. AP-credit entrants with thinner foundations than their placement implied feel it most.
How do I study for 21-122 exams?
Mixed integral sets so technique selection becomes automatic (that choice is the exam skill), and a convergence-test decision chart practiced until classification is rapid and justified. Both under time limits before each exam.
Why is the series unit in 21-122 so hard?
It's the first calculus topic that's more proof than procedure: deciding whether an infinite sum converges means choosing a test, verifying its conditions, and arguing the conclusion. That inverts the drill-based studying that worked for integration.
More CMU courses
21-120: Differential and Integral Calculus
21-120 is CMU's first calculus course: limits, derivatives, applications of differentiation, and the integral through the fundamental theorem. It's the standard entry point for students not placing into 21-122 via AP credit, and it feeds every quantitative major on campus.
21-127: Concepts of Mathematics
21-127 is CMU's introduction to mathematical proof: logic, sets, functions, induction, number theory, and combinatorics. It's required early for CS and math majors as the gateway to everything proof-based, including 15-251, and for most students it's the first course where the answer is an argument.
21-241: Matrices and Linear Transformations
21-241 is CMU's first linear algebra course: systems and matrices, vector spaces, linear transformations, determinants, eigenvalues, and orthogonality. It serves CS, science, and engineering students and blends computation with proof more than most schools' equivalents.
21-259: Calculus in Three Dimensions
21-259 is CMU's multivariable calculus course: vectors and surfaces, partial derivatives, multiple integrals, and the vector calculus capstone of line integrals, Green's, Stokes', and the divergence theorem. It's required across engineering and the sciences.