# York MATH 1310 Integral Calculus: Topics, Guide and Practice

> A study guide for York University MATH 1310 (Integral Calculus with Applications): integration methods, volumes, improper integrals and series.

Source: https://practicecalculus.com/courses/york-university-math-1310
Updated: 2026-11-08

## At a glance

- **Course code:** SC/MATH 1310 3.00
- **Credits:** 3.00
- **Prerequisites:** One of SC/MATH 1013, SC/MATH 1300, GL/MATH 1901, SC/ISCI 1401; for non-science students only, six credits from MATH 1530 and 1540, MATH 1550, ECON 1530 and 1540
- **Exclusions:** SC/MATH 1014, 1505, GL/MATH/MODR 1940, SC/ISCI 1402, 1410
- **Covers:** Exponential and log functions defined by integrals, methods of integration, volumes, approximate areas, distance travelled, improper integrals, infinite series
- **Next course:** MATH 2015 or MATH 2310 (multivariable calculus)
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- MATH 1310 is the 3-credit second course in York's theoretical calculus series, designed to follow MATH 1300.
- It defines exponential and logarithmic functions properly using integrals.
- It covers methods of integration, volumes, approximate areas, distance travelled and improper integrals.
- It gives a precise definition of infinite sums and studies their applications.

**MATH 1310** is Integral Calculus with Applications at York University, the second course in a series designed to follow MATH 1300. Like MATH 1300, it is theoretical: it gives precise definitions and proves key results, while still covering the standard integration techniques and applications. This guide maps each MATH 1310 topic to free practice so you can make the computations second nature.

## What York MATH 1310 covers

York's Department of Mathematics and Statistics describes MATH 1310 as continuing the study of integral calculus begun in MATH 1300. Exponential and logarithmic functions are properly defined using integrals. Methods of integration are discussed. Applications are made to compute volumes, approximate areas and distance travelled. Improper integrals are used to study unbounded areas. A precise definition for adding up infinitely many numbers is given, and some of its many applications are examined. The department notes that it is a theoretical course emphasizing precise definitions and featuring proofs of some important results.

This maps to the core of a standard **Calculus 2** course. Parametric curves, polar coordinates and differential equations are not mentioned in the description.

## MATH 1310 topic map

| MATH 1310 topic | Practice | Read |
| --- | --- | --- |
| The integral and FTC review | [FTC part 1](/learn/calculus-1/integrals/fundamental-theorem-part-1), [accumulation functions](/learn/calculus-1/integrals/accumulation-functions) | [FTC guide](/blog/fundamental-theorem-of-calculus) |
| Logarithm defined by an integral | [Exponential and log derivatives](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Natural logarithm](/functions/natural-logarithm), [integral of 1/x](/integrals/1-over-x) |
| Exponential functions | [Exponentials and logs](/learn/calculus-1/functions-warmup/exponentials-and-logs) | [Natural exponential function](/functions/natural-exponential-function) |
| Substitution | [u-substitution refresher](/learn/calculus-2/integration-techniques/u-substitution-refresher) | [u-substitution guide](/blog/u-substitution) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts guide](/blog/integration-by-parts) |
| Trig methods and partial fractions | [Trig substitution](/learn/calculus-2/integration-techniques/trig-substitution), [partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions guide](/blog/partial-fractions) |
| Volumes | [Disks](/learn/calculus-1/applications-of-integrals/volumes-disks), [washers](/learn/calculus-1/applications-of-integrals/volumes-washers) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Approximate areas | [Riemann sums](/learn/calculus-1/integrals/riemann-sums), [trapezoid sums](/learn/calculus-1/integrals/trapezoid-sums) | [Riemann sums guide](/blog/riemann-sums) |
| Distance travelled | [Net change](/learn/calculus-1/applications-of-integrals/net-change) | |
| Improper integrals | [Infinite limits of integration](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration), [comparison test](/learn/calculus-2/improper-integrals-and-applications/comparison-test) | [Improper integrals guide](/blog/improper-integrals) |
| Infinite series | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums), [choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests), [geometric series](/blog/geometric-series) |

## MATH 1310 vs MATH 1014

| Course | Title | Style |
| --- | --- | --- |
| MATH 1310 | Integral Calculus with Applications | Theoretical, follows MATH 1300 |
| MATH 1014 | Applied Calculus II | Applied integral calculus with an intro to sequences and series |

They exclude each other for credit. Before MATH 1310 most students take [MATH 1300](/courses/york-university-math-1300), and a common next step for science and engineering is [MATH 2015](/courses/york-university-math-2015).

## Example problems at the MATH 1310 level

**1. The logarithm as an integral.** Defining $\ln x = \int_1^x \frac{dt}{t}$, the Fundamental Theorem gives $\frac{d}{dx}\ln x = \frac{1}{x}$ immediately.

**2. Integration by parts.**

$$\int \ln x\,dx = x\ln x - \int x\cdot\frac{1}{x}\,dx = x\ln x - x + C$$

**3. Distance travelled.** If $v(t) = 3t^2$ m/s, the distance in the first $2$ seconds is $\int_0^2 3t^2\,dt = 8$ m.

**4. A telescoping series.**

$$\sum_{n=1}^{\infty}\frac{1}{n(n + 1)} = \sum_{n=1}^{\infty}\left(\frac{1}{n} - \frac{1}{n + 1}\right) = 1$$

## How to do well in MATH 1310

- **Understand the definitions behind the formulas.** If $\ln x$ is defined as an integral, its properties are theorems you can prove. Expect questions that ask why, not just what.
- **Practise techniques in mixed sets.** Recognising the right method is the skill that takes longest. Try [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy).
- **Treat series carefully.** A series is the limit of its partial sums. Write the partial sum when you can, especially for telescoping series.
- **Draw before computing volumes.** A sketch shows the radius and the limits of integration.

> **Tip:** an improper integral is a limit. Replace the problem bound with a variable, integrate, then take the limit.

## Frequently asked questions

### What is MATH 1310 at York?

MATH 1310 is Integral Calculus with Applications. York's department describes it as continuing the integral calculus begun in MATH 1300: exponential and logarithmic functions are defined using integrals, methods of integration are discussed, applications include volumes, approximate areas and distance travelled, improper integrals are used to study unbounded areas, and a precise definition of adding infinitely many numbers is given.

### MATH 1310 vs MATH 1014?

They exclude each other for credit. MATH 1310 is theoretical, with definitions and proofs, and is recommended for math majors and computer science students. MATH 1014, Applied Calculus II, covers the theory and applications of integral calculus and ends with an introduction to sequences and series.

### What do I need before MATH 1310?

One of SC/MATH 1013, SC/MATH 1300, GL/MATH 1901 or SC/ISCI 1401. Non-science students can also qualify with certain six-credit combinations listed by the department.

### What comes after MATH 1310?

MATH 1310 is a prerequisite option for multivariable courses such as MATH 2015 (Applied Multivariate and Vector Calculus), and the department says it prepares students for courses like MATH 2001.