Virginia Tech MATH 1226 Calculus 2: Topics, Study Guide, Practice

MATH 1226 is the second semester of Calculus of a Single Variable at Virginia Tech, taken after MATH 1225 by engineering, science and math students. It is a 4-credit course in the Pathways Quantitative and Computational Thinking curriculum, with an emphasis on software-based techniques alongside hand computation. This guide maps every MATH 1226 topic to free practice, so you can learn each method by working problems with instant feedback.

What MATH 1226 covers

The Virginia Tech math department describes MATH 1226 as covering integration techniques and applications, numerical integration, improper integrals, sequences and series, power series, parametric curves and polar coordinates. Its unit list follows the Stewart textbook chapters:

1. Integration and applications: substitution and the definite integral (review), area between curves, volumes, work, average value, integration by parts, trig integrals, trig substitution.
2. More techniques and applications: partial fractions, strategy, approximate integration, the Gini index, centers of mass, indeterminate forms, improper integrals, probability.
3. Sequences and series: convergence tests, power series, Taylor and Maclaurin series, applications of Taylor polynomials.
4. Parametric and polar: parametric curves, calculus with parametric curves, polar coordinates.

That lines up with a standard Calculus 2 course, with a few more applied topics than usual.

MATH 1226 topic map

MATH 1226 vs MATH 1225 vs MATH 2204

Applications like volumes, work and average value appear in Unit 1 of MATH 1226 rather than at the end of MATH 1225. Review the MATH 1225 guide for derivatives and substitution, and see the MATH 2204 guide for what comes next.

Example problems at the MATH 1226 level

1. Trig substitution. Evaluate \int \frac{dx}{\sqrt{9 - x^2}}. Let x = 3\sin\theta, so dx = 3\cos\theta\,d\theta and \sqrt{9 - x^2} = 3\cos\theta:

2. An alternating series. \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} converges by the alternating series test, since \frac{1}{n} decreases to 0. It does not converge absolutely, because the harmonic series diverges.

3. Radius of convergence. For \sum_{n=0}^{\infty} \frac{x^n}{2^n}, the ratio test gives

so the radius of convergence is R = 2.

How to do well in MATH 1226

• Practice choosing a technique, not just executing one. Exams mix integrals together. The choosing a strategy and techniques practice lessons train that skill.
• Learn the series tests in order. Nth-term test first, then geometric and p-series, then comparison, alternating and ratio tests. The ratio test page shows when it is the right call.
• Know your Maclaurin series by heart. Keep e^x, \sin x, \cos x and \frac{1}{1-x} on your formula sheet until you can write them from memory.
• Avoid the usual errors listed in common calculus mistakes, such as dropping the dx conversion in substitutions.
• Do timed mixed sets in practice before every unit exam.

Tip: before using any series test, check \lim a_n. If it is not 0, the series diverges by the nth-term test and you are done.