UCSD MATH 20B Calculus: Topics, Study Guide and Practice

MATH 20B is the second quarter of UC San Diego's calculus sequence for science and engineering. It picks up where MATH 20A stops, at the integral, and carries students through integration methods, infinite series and polar coordinates. This guide maps every MATH 20B topic to free practice so you can learn each technique by working problems with instant feedback.

What UCSD MATH 20B covers

According to the UC San Diego catalog, MATH 20B is integral calculus of one variable and its applications. It works with exponential, logarithmic, hyperbolic and trigonometric functions, develops methods of integration, and then turns to infinite series, polar coordinates in the plane and complex exponentials.

At most universities this material is split between the end of Calculus 1 (the definite integral and the Fundamental Theorem) and Calculus 2 (techniques, series and polar coordinates). MATH 20B covers both pieces in one quarter, and adds complex exponentials such as e^{i\theta} = \cos\theta + i\sin\theta.

MATH 20B topic map

Each row links to a lesson you can practice and to a reference page or guide that explains the idea.

MATH 20B vs MATH 20A and MATH 20C

MATH 20B is where integration becomes a toolbox rather than a single rule. Series also matter later: Taylor expansions return in differential equations and physics. When you finish, the MATH 20C guide covers the move to several variables.

Example problems at the MATH 20B level

1. Integration by parts. Evaluate \int x\cos x\,dx with u = x and dv = \cos x\,dx, so du = dx and v = \sin x:

2. A geometric series. Find the sum of \sum_{n=0}^{\infty} \frac{3}{4^n}.

This is geometric with first term a = 3 and ratio r = \frac{1}{4}, and |r| < 1, so

3. A complex exponential. Write e^{i\pi/2} in the form a + bi.

By Euler's formula, e^{i\pi/2} = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} = 0 + i = i.

How to do well in UCSD MATH 20B

• Practice choosing the method. Exams rarely tell you which technique to use. The choosing a strategy lesson and mixed practice exams train that decision.
• Memorize the core integrals. The calculus formula sheet lists every basic integral and series you need.
• Learn the geometric series cold. Most series arguments compare to it, so start with the geometric series formula.
• Check answers by differentiating. Differentiate your antiderivative to catch sign errors, one of the common calculus mistakes that costs the most points.

Tip: for series, ask the same three questions every time: does the term go to zero, does it look like a geometric or p-series, and does the ratio test apply?