York MATH 1310 Integral Calculus: Topics, Guide and Practice

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 vs MATH 1014

They exclude each other for credit. Before MATH 1310 most students take MATH 1300, and a common next step for science and engineering is 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.

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.

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.
• 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.