# Waterloo MATH 138 Calculus 2: Topics, Study Guide and Practice

> A study guide for MATH 138 at the University of Waterloo: integration, improper integrals, differential equations, series and Taylor's theorem, with practice.

Source: https://practicecalculus.com/courses/university-of-waterloo-math-138
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 138
- **Units:** 0.50
- **Prerequisites:** At least 60% in MATH 137, or MATH 147, or at least 70% in MATH 116, 117 or 127
- **Antirequisites:** MATH 118, 119, 128 and 148
- **Covers:** Riemann integral, Fundamental Theorem of Calculus, integration methods, improper integrals, linear and separable differential equations, series convergence, Taylor's theorem, Big-O notation, power and Taylor series
- **Next course:** MATH 237 (Calculus 3 for Honours Mathematics)
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- MATH 138 is Calculus 2 for Honours Mathematics at Waterloo, worth 0.50 units, and follows MATH 137.
- It starts with the Riemann integral and the Fundamental Theorem of Calculus, so integration first appears here.
- It covers linear and separable differential equations, convergence tests for series, and Taylor's theorem with Big-O notation.
- It ends with binomial series, power series and Taylor series.

**MATH 138** is Calculus 2 for Honours Mathematics at the University of Waterloo, the course that follows [MATH 137](/courses/university-of-waterloo-math-137). Because MATH 137 is entirely differential calculus, MATH 138 is where Waterloo math students meet integration, and it then moves through differential equations, series and Taylor's theorem in one term. This guide maps each topic to practice so you can work through it one problem at a time.

## What Waterloo MATH 138 covers

The Waterloo academic calendar lists the topics as the Riemann integral and approximations, antiderivatives and the Fundamental Theorem of Calculus, change of variables and methods of integration, applications of the integral, improper integrals, linear and separable differential equations with applications, convergence tests for series, Taylor polynomials and Taylor's theorem, Big-O notation, the binomial series, and functions defined as power series and Taylor series. Some topics are illustrated with computer software.

Measured against a standard sequence, that is the integral half of **Calculus 1** plus most of **Calculus 2**: techniques, improper integrals, differential equations and series. Parametric curves and polar coordinates are not on the list.

## MATH 138 topic map

| MATH 138 topic | Practice | Read |
| --- | --- | --- |
| Riemann integral and approximations | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| Antiderivatives and the FTC | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Antiderivatives](/blog/antiderivatives), [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1) |
| Change of variables | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [u-substitution](/blog/u-substitution) |
| Methods of integration | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts), [partial fractions](/blog/partial-fractions), [trig substitution](/blog/trig-substitution) |
| Applications of the integral | [Volumes with washers](/learn/calculus-1/applications-of-integrals/volumes-washers) | [Area between curves](/blog/area-between-curves), [volumes of revolution](/blog/volumes-of-revolution) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Separable differential equations | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Natural exponential function](/functions/natural-exponential-function) |
| Convergence tests for series | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests), [ratio test](/rules/ratio-test) |
| Taylor polynomials and Taylor's theorem | [Estimating with Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/estimating-with-taylor) | [Taylor's theorem](/theorems/taylors-theorem) |
| Power series and Taylor series | [Common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series), [power series](/blog/power-series) |

## MATH 138 in Waterloo's calculus sequence

| Course | Title | Prerequisite |
| --- | --- | --- |
| [MATH 137](/courses/university-of-waterloo-math-137) | Calculus 1 for Honours Mathematics | 4U Calculus and Vectors |
| MATH 138 | Calculus 2 for Honours Mathematics | 60% in MATH 137, or MATH 147, or 70% in MATH 116, 117 or 127 |
| MATH 148 | Calculus 2 (Advanced Level) | The advanced-level alternative to MATH 138 |
| [MATH 237](/courses/university-of-waterloo-math-237) | Calculus 3 for Honours Mathematics | 60% in MATH 138 (or MATH 148, or 70% in MATH 128) plus linear algebra |

MATH 138 is an antirequisite of MATH 118, 119, 128 and 148, so students take only one second calculus course.

## Example problems at the MATH 138 level

**1. A linear differential equation.** Solve $y' + y = e^x$. Multiply by the integrating factor $e^x$:

$$(e^x y)' = e^{2x} \quad\Rightarrow\quad e^x y = \frac{1}{2}e^{2x} + C \quad\Rightarrow\quad y = \frac{1}{2}e^x + Ce^{-x}$$

**2. Big-O and Taylor's theorem.** Since $\cos x = 1 - \frac{x^2}{2} + O(x^4)$ as $x \to 0$,

$$\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \lim_{x \to 0} \frac{\frac{x^2}{2} + O(x^4)}{x^2} = \frac{1}{2}$$

**3. The binomial series.** For $|x| < 1$,

$$\sqrt{1 + x} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \cdots$$

so $\sqrt{1.02} \approx 1 + 0.01 - 0.00005 = 1.00995$.

## How to do well in MATH 138

- **Drill integration until you can choose the method fast.** The [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) lesson and daily [practice](/practice) sets build that instinct.
- **Memorize the core series.** Knowing $e^x$, $\sin x$, $\cos x$, $\frac{1}{1-x}$ and the binomial series by heart makes Big-O questions quick.
- **Check every antiderivative.** Differentiate your answer. Our [common calculus mistakes](/blog/common-calculus-mistakes) guide lists the slips that cost the most marks.
- **Use one reference.** The [calculus formula sheet](/formulas) keeps integrals and series together.

> **Tip:** Big-O notation is a bookkeeping tool. Write each Taylor expansion only to the order the limit needs, and drop everything smaller into $O(\cdot)$.

## Frequently asked questions

### What is MATH 138 at Waterloo?

MATH 138 is Calculus 2 for Honours Mathematics. It introduces the Riemann integral and the Fundamental Theorem of Calculus, then covers integration methods and applications, improper integrals, differential equations, series and Taylor series.

### What do I need to take MATH 138?

At least 60% in MATH 137, or credit for MATH 147, or at least 70% in one of MATH 116, 117 or 127.

### Is MATH 138 harder than MATH 137?

Many students find it harder because it combines a full course of integration with series and Taylor's theorem. Integration takes strategy, and series need careful reasoning about convergence, so plan for regular practice from the first week.

### What comes after MATH 138?

Honours Mathematics students continue to MATH 237, Calculus 3 for Honours Mathematics, which also requires a linear algebra course such as MATH 136.

### How should I prepare for MATH 138?

Make sure derivatives are automatic and review antiderivatives of basic functions. Practicing u-substitution before the term begins gives you a head start on integration methods.