# Northwestern MATH 220-2 Integral Calculus: Topics and Practice > A study guide for Northwestern MATH 220-2 Single-Variable Integral Calculus: the Fundamental Theorem, techniques, improper integrals and first-order ODEs. Source: https://practicecalculus.com/courses/northwestern-math-220-2 Updated: 2026-11-08 ## At a glance - **Course code:** MATH 220-2 - **Units:** 1 unit - **Offered:** Not stated on the First-Year Focus page - **Prerequisites:** MATH 218-2, MATH 220-0 (former) or MATH 220-1 - **Covers:** Definite integrals, antiderivatives and the Fundamental Theorem, transcendental and inverse functions, areas and volumes, techniques of integration, numerical integration, improper integrals, and first-order linear and separable ODEs - **Next course:** MATH 226-0 (Sequences and Series) or MATH 230-1 (Multivariable Differential Calculus) - **Practice here:** [Calculus 1](/learn/calculus-1) integrals units and [Calculus 2](/learn/calculus-2) ## Key takeaways - MATH 220-2 is the second quarter of Northwestern's standard calculus sequence and is all about integrals. - It runs from the definition of the integral and the Fundamental Theorem through integration techniques and improper integrals. - It ends with first-order linear and separable differential equations. - Series are not in 220-2; they are in MATH 226-0, which math and statistics majors must take. **Northwestern MATH 220-2**, Single-Variable Integral Calculus, is the second course in Northwestern's standard calculus sequence. Where [MATH 220-1](/courses/northwestern-math-220-1) is all about derivatives, 220-2 is all about integrals: what they mean, how to compute them and how to use them. This guide maps each topic to free practice with instant feedback. ## What MATH 220-2 covers Northwestern's course description lists: definite integrals, antiderivatives, and the fundamental theorem of calculus; transcendental and inverse functions; areas and volumes; techniques of integration, numerical integration, and improper integrals; and first-order linear and separable ordinary differential equations. The course is worth one unit, and the prerequisite is MATH 218-2, MATH 220-0 (former) or MATH 220-1. In semester terms, MATH 220-2 covers the integration half of **Calculus 1** plus the integration techniques and differential equations of **Calculus 2**. The other part of a typical Calculus 2 course, series, lives in a separate course, MATH 226-0. ## MATH 220-2 topic map | MATH 220-2 topic | Practice | Read | | --- | --- | --- | | Definite integrals | [Riemann sums](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums guide](/blog/riemann-sums) | | Antiderivatives | [Reverse power rule](/learn/calculus-1/integrals/reverse-power-rule) | [Antiderivatives](/blog/antiderivatives) | | Fundamental Theorem of Calculus | [Fundamental Theorem, part 2](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [FTC part 2](/theorems/fundamental-theorem-of-calculus-part-2) | | Transcendental and inverse functions | [Inverse trig derivatives](/learn/calculus-1/derivative-rules/inverse-trig-derivatives) | [Integral of 1/(1+x²)](/integrals/1-over-1-plus-x-squared) | | Areas | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves guide](/blog/area-between-curves) | | Volumes | [Volumes with disks](/learn/calculus-1/applications-of-integrals/volumes-disks) | [Volumes of revolution](/blog/volumes-of-revolution) | | 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) | | Trig substitution and partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Trig substitution](/blog/trig-substitution) | | Numerical integration | [Trapezoid sums](/learn/calculus-1/integrals/trapezoid-sums) | [Riemann sums practice](/practice-problems/riemann-sums) | | Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) | | Separable ODEs | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Differential equations practice](/practice-problems/differential-equations) | | Growth models | [Growth and decay](/learn/calculus-2/improper-integrals-and-applications/growth-and-decay) | [Natural exponential function](/functions/natural-exponential-function) | First-order linear equations, solved with an integrating factor, are also on the list. They build directly on the integration techniques above. ## Where 220-2 fits in the sequence The department places students with a 4 on AP Calculus AB (or the AB subscore of BC) into MATH 220-2. After 220-2, students outside the McCormick School of Engineering usually take MATH 230-1 for multivariable calculus, and engineering students take MATH 228-1. MATH 226-0, Sequences and Series, requires 220-2 or 218-3 and is required for mathematics and statistics majors and minors, but the department notes it is not required for most science and engineering majors. ## Example problems at the MATH 220-2 level **1. An inverse trig integral.** $$\int_0^1 \frac{dx}{1 + x^2} = \arctan 1 - \arctan 0 = \frac{\pi}{4}$$ **2. Partial fractions.** $$\int \frac{dx}{x^2 - 1} = \frac{1}{2}\ln\left|\frac{x - 1}{x + 1}\right| + C$$ **3. A first-order linear ODE.** For $y' + y = 1$, the integrating factor $e^x$ gives $(ye^x)' = e^x$, so $y = 1 + Ce^{-x}$. ## How to do well in MATH 220-2 - **Understand the Fundamental Theorem deeply.** Almost every later topic depends on it, so work through [both parts](/learn/calculus-1/integrals/fundamental-theorem-part-1). - **Build a technique checklist.** Practice [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) until you can name the method in seconds. - **Sketch areas and solids.** A quick drawing prevents most setup mistakes in area and volume problems. - **Use the quarter rhythm.** Ten weeks is short, so daily practice beats weekend cramming. Next, see the [MATH 230-1 guide](/courses/northwestern-math-230-1) for multivariable calculus. ## Frequently asked questions ### What is Northwestern MATH 220-2? MATH 220-2, Single-Variable Integral Calculus, is a one-unit course covering definite integrals, antiderivatives and the Fundamental Theorem, transcendental and inverse functions, areas and volumes, techniques of integration, numerical integration, improper integrals, and first-order linear and separable differential equations. ### What are the prerequisites for MATH 220-2? MATH 218-2, MATH 220-0 (former) or MATH 220-1. ### Which AP score places into MATH 220-2? The department recommends MATH 220-2 for a 4 on AP Calculus AB (or the AB subscore of the BC exam). ### Does MATH 220-2 cover infinite series? No. Series, power series and Taylor series are in MATH 226-0, Sequences and Series, which requires MATH 218-3 or 220-2. ### What comes after MATH 220-2? MATH 230-1, Multivariable Differential Calculus, for students outside the engineering school, MATH 228-1 for McCormick engineering students, and MATH 226-0 for math and statistics majors and some others.