MATH 141: Calculus II

MATH 141 at the University of Maryland Global Campus is a four-credit continuation of MATH 140.

UMGC describes a study of integration and functions, with a focus on techniques of integration, improper integrals, applications such as volumes, work, arc length and moments, inverse, exponential and logarithmic functions, and sequences and series.

The course asks you to model and solve problems with integrals and to use infinite series to approximate functions. Choosing among techniques is the main skill, followed by testing whether series converge.

What MATH 141 Covers

Key Concepts Explained

Improper Integrals

Replace the infinite limit with a variable, integrate, then take a limit.

Volumes by Discs

For y = square root of x rotated about the x-axis from 0 to 4, volume = pi times the integral of x dx from 0 to 4 = pi x 8 = 8 pi.

Series Tests

The ratio test compares consecutive terms. If the limit of the ratio is less than 1 the series converges; greater than 1, it diverges; equal to 1, it is inconclusive.

Typical Assignments and How to Approach Them

Setting up Application Problems

Most lost marks in applications come from the setup. Draw the region, mark the slice, state its dimensions in terms of one variable, and write the integral before you evaluate it. Include units in the final statement.

A Series Checklist

• Do the terms approach zero? If not, the series diverges.
• Does the pattern match a known type, such as geometric or p-series?
• Which test fits, and are its conditions satisfied?
• State the conclusion and name the test used.

More Worked Examples

Integration by parts: the integral of ln x dx. Let u = ln x and dv = dx, so du = dx/x and v = x. The result is x ln x - the integral of 1 dx, which is x ln x - x + C.

Ratio test: for the series of n/2^n, the ratio of consecutive terms is (n + 1)/(2n), which tends to 1/2. Since 1/2 is less than 1, the series converges.

Applications in One Table

UMGC lists volumes, work, arc length and moments. Each has a representative slice and an integral that adds them up.

A Short Self-Test

Try these without notes to find your weak spots early.

• Integrate x e^x dx by parts.
• Decide whether the integral of 1/x from 1 to infinity converges.
• Set up the volume of the solid formed by rotating y = x^2 from 0 to 1 about the x-axis.
• Test the series with terms 1/n^2 for convergence.
• Find the first three terms of the geometric series with a = 3 and r = 1/2.

Check each answer against your lesson notes, and write the name of the technique beside every problem.

Where Students Get Stuck

• Wrong technique. Classify first.
• Setup of volumes. Decide on discs, washers or shells and justify.
• Series tests. Check conditions before concluding.
• Missing limits. Show the limit process in improper integrals.

Study Tips for MATH 141

• Keep a technique sheet with an example for each.
• Always sketch the region.
• Check integrals by differentiating.
• Practice mixed series problems.

How We Help with MATH 141

Send the problem set, your notes and your working. A maths tutor can explain each method, prepare fully worked solutions to study from, or check your steps.

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