# U of T MAT136H1 Calculus II: Topics, Study Guide and Practice

> A study guide for MAT136H1 at the University of Toronto: integration, improper integrals, Taylor series and differential equations, with free practice.

Source: https://practicecalculus.com/courses/university-of-toronto-mat136h1
Updated: 2026-11-03

## At a glance

- **Course code:** MAT136H1 (Faculty of Arts and Science, St. George)
- **Hours:** 36 lecture hours and 12 tutorial hours
- **Prerequisites:** One of MAT135H1, MAT130H1, MAT186H1, MAT196H1, APS162H1, ESC194H1 or an equivalent first calculus course
- **Breadth:** The Physical and Mathematical Universes (5)
- **Covers:** Fundamental Theorem of Calculus, substitution and integration by parts, improper integrals, Taylor polynomials and series, ratio test, radius of convergence, first-order differential equations
- **Practice here:** [Calculus 1 integrals](/learn/calculus-1/integrals/undoing-derivatives) and [Calculus 2](/learn/calculus-2)

## Key takeaways

- MAT136H1 is the second half of U of T's standard science calculus sequence, taken after MAT135H1.
- It is an integral calculus course that starts from the Fundamental Theorem of Calculus.
- It also covers Taylor series, the ratio test and radius of convergence, plus first-order differential equations and systems.
- Computer algebra systems appear in the course for integration and for solving differential equations.

**MAT136H1** is Calculus II at the University of Toronto and the second course in the default calculus sequence for science students, right after [MAT135H1](/courses/university-of-toronto-mat135h1). Where MAT135H1 is about derivatives, MAT136H1 is about integrals and what you can do with them, from areas and totals to infinite series and differential equations. This guide maps every MAT136H1 topic to practice so you can learn by solving problems.

## What U of T MAT136H1 covers

According to the Arts and Science calendar, MAT136H1 opens with the Fundamental Theorem of Calculus, which ties accumulating change to finding area. From there students use integrals to compute areas, volumes and totals, to analyze equations involving derivatives, to make sense of infinite sums, and to write complicated functions as "infinite polynomials".

The listed topics are integration by substitution and by parts, improper integrals, computer algebra systems for integration, Taylor polynomials and Taylor series, the ratio test and radius of convergence, and first-order differential equations and systems, including separable equations and modelling. That matches the integral half of a standard **Calculus 1** course plus the core of **Calculus 2**, without trig substitution, partial fractions, parametric curves or polar coordinates in the listed topics.

## MAT136H1 topic map

| MAT136H1 topic | Practice | Read |
| --- | --- | --- |
| Riemann sums and the definite integral | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| Fundamental Theorem of Calculus | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus), [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1) |
| Substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [u-substitution](/blog/u-substitution), [u-substitution rule](/rules/u-substitution-rule) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts) |
| Areas, volumes and totals | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [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) |
| Taylor polynomials | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| Ratio test and radius of convergence | [Power series and radius of convergence](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Ratio test](/rules/ratio-test), [power series](/blog/power-series) |
| Separable differential equations | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Natural exponential function](/functions/natural-exponential-function) |

## MAT136H1 vs MAT149H1

| Course | Prerequisite | Style |
| --- | --- | --- |
| MAT136H1 | MAT135H1 or an equivalent first calculus course | Applied, modelling, uses computer algebra |
| [MAT149H1](/courses/university-of-toronto-mat149h1) | MAT148H1, or MAT135H1 or MAT130H1 with 70% or higher | Proof-based, Taylor's theorem and series |

Both count as a second calculus course for programs that ask for MAT135H1 and MAT136H1. If you later want proof-based multivariable calculus in [MAT237Y1](/courses/university-of-toronto-mat237y1), the calendar pairs MAT135H1 and MAT136H1 with a proofs course such as MAT138H1, plus linear algebra.

## Example problems at the MAT136H1 level

**1. Integration by parts.** Evaluate $\int x\cos x\,dx$ with $u = x$, $dv = \cos x\,dx$:

$$\int x\cos x\,dx = x\sin x - \int \sin x\,dx = x\sin x + \cos x + C$$

**2. A Taylor polynomial.** The degree 3 Taylor polynomial of $e^x$ at $0$ is

$$T_3(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$$

so $e^{0.1} \approx 1 + 0.1 + 0.005 + 0.000167 \approx 1.10517$.

**3. A separable differential equation.** Solve $\dfrac{dy}{dt} = 2ty$ with $y(0) = 3$. Separate and integrate:

$$\int \frac{dy}{y} = \int 2t\,dt \quad\Rightarrow\quad \ln|y| = t^2 + C$$

so $y = Ae^{t^2}$, and $y(0) = 3$ gives $y = 3e^{t^2}$.

## How to do well in MAT136H1

- **Learn to pick the method.** Most integration mistakes come from choosing the wrong technique. Our [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) lesson trains exactly that decision.
- **Treat series as pattern recognition.** Memorize the common Maclaurin series and the [ratio test](/rules/ratio-test), then read the [convergence tests guide](/blog/convergence-tests).
- **Use the formula sheet.** The [calculus formula sheet](/formulas) lists the integrals and series you will use most.
- **Practice under time pressure.** Mixed sets in [practice](/practice) with fresh numbers each time get you ready for term tests, and [how to prepare for a calculus exam](/blog/how-to-prepare-for-a-calculus-exam) has a study plan.

> **Tip:** check every antiderivative by differentiating it. It takes seconds and catches most sign errors before they cost marks.

## Frequently asked questions

### What is MAT136H1 at U of T?

MAT136H1 is Calculus II in the Faculty of Arts and Science. It covers integration and its applications, improper integrals, Taylor polynomials and series, and first-order differential equations, with an emphasis on modelling.

### What do I need before MAT136H1?

You need a first calculus course: usually MAT135H1, or an equivalent such as MAT130H1, a UTM or UTSC first calculus course, or an engineering first-year calculus course listed in the calendar.

### Is MAT136H1 harder than MAT135H1?

Many students find it harder, because integration takes more strategy than differentiation and series are a new way of thinking. Practicing integration techniques until you can recognize which one fits is the best defence.

### MAT136H1 vs MAT149H1: which should I take?

MAT136H1 is applied and focuses on computing and modelling. MAT149H1, Calculus II with Proofs, covers similar calculus with a theoretical, proof-based approach. Students from MAT135H1 need 70% or higher to enter MAT149H1.

### How should I prepare for MAT136H1?

Make sure derivatives are automatic, since every integral is a derivative in reverse. Then practice antiderivatives and u-substitution before the term starts.