UW-Madison MATH 222 Calculus 2: Topics, Study Guide and Practice

MATH 222 is the second semester of Calculus and Analytic Geometry at the University of Wisconsin-Madison. Students take it after MATH 221 on the way to MATH 234, and it is required for most engineering, physics and math programs. This guide maps every MATH 222 topic to free practice, so you can learn each technique by working problems with instant feedback.

What MATH 222 covers

The UW-Madison Guide lists six themes for MATH 222: techniques of integration, improper integrals, first-order ordinary differential equations, sequences and series, Taylor series, and vector geometry in two and three dimensions.

The first five match a standard Calculus 2 course. The vector geometry at the end (vectors, dot and cross products, lines and planes) is material that many schools teach at the start of Calculus 3, so MATH 234 can move straight into functions of several variables.

MATH 222 topic map

MATH 222 in the sequence

MATH 213 also touches series and differential equations, but it is a survey course aimed at business and social science applications and does not lead to MATH 234. For the full path, see the MATH 221 guide and the MATH 234 guide.

Example problems at the MATH 222 level

1. A separable differential equation. Solve \frac{dy}{dx} = 2xy with y(0) = 3. Separate: \frac{dy}{y} = 2x\,dx, so \ln|y| = x^2 + C and y = Ae^{x^2}. The initial condition gives A = 3:

2. A series test. Does \sum_{n=1}^{\infty} \frac{n}{3^n} converge? Use the ratio test:

so the series converges.

3. A cross product. For \mathbf{u} = \langle 1, 0, 0 \rangle and \mathbf{v} = \langle 0, 1, 1 \rangle, \mathbf{u} \times \mathbf{v} = \langle 0 \cdot 1 - 0 \cdot 1,\ 0 \cdot 0 - 1 \cdot 1,\ 1 \cdot 1 - 0 \cdot 0 \rangle = \langle 0, -1, 1 \rangle, which is perpendicular to both.

How to do well in MATH 222

• Build an integration decision habit. Before integrating, ask: substitution, parts, trig, or partial fractions? The choosing a strategy lesson drills that first step.
• Treat series as a checklist. Try the nth-term test first, then compare to a geometric or p-series, then reach for the ratio test. Practice the order until it is automatic.
• Memorize five Maclaurin series. e^x, \sin x, \cos x, \frac{1}{1-x} and \ln(1+x) answer a surprising share of Taylor questions. Keep them on your formula sheet.
• Review the usual traps in common calculus mistakes, especially forgetting +C and losing signs in parts.
• Practice under exam conditions with mixed sets in practice, using the plan in how to prepare for a calculus exam.

Tip: the geometric series formula is the backbone of MATH 222 series. Power series, comparison tests and Taylor series all lean on it.