MTH1020: Applications of Integration Cheat Sheet

This cheat sheet summarizes key applications of integration, including calculating areas between curves, average function values, volumes of solids of revolution (using discs and cylindrical shells), lengths of curves, and surface areas of revolution. It provides formulas, methods, and examples for each topic.

Core Principles

• Integration is used to find areas, volumes, lengths, and surface areas that cannot be calculated with simple geometric formulas.
• The method of discs and cylindrical shells are common techniques for finding volumes of revolution.
• The arc length formula involves integrating the square root of 1 plus the square of the derivative.
• Surface area of revolution formulas depend on the axis of rotation and the function's form (Cartesian, parametric).

Action Steps

• Identify the region or curve to be analyzed.
• Determine the appropriate method (discs, shells, arc length, surface area).
• Set up the integral with the correct limits of integration.
• Evaluate the integral to find the desired quantity (area, volume, length, surface area).

Formulas

• Area between curves: $A = \int_a^b |f(x) - g(x)| dx$
• Average value of a function: $f_{avg} = \frac{1}{b-a} \int_a^b f(x) dx$
• Volume (Discs): $V = \int_a^b \pi [f(x)]^2 dx$
• Volume (Cylindrical Shells): $V = \int_a^b 2\pi x f(x) dx$
• Arc Length: $L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$
• Surface Area (about x-axis): $A = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dx}{dy}\right)^2} dy$ or $A = \int_a^b 2\pi f(x) \sqrt{1 + \left(f'(x)\right)^2} dx$
• Surface Area (parametric): $A = \int_a^b 2\pi y(t) \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

Key Terms

• Riemann Sum: A method to approximate the area under a curve by dividing it into rectangles.
• Solid of Revolution: A 3D shape formed by rotating a 2D shape around an axis.
• Method of Discs: A technique for finding volumes of revolution by integrating the areas of circular discs.
• Method of Cylindrical Shells: A technique for finding volumes of revolution by integrating the surface areas of cylindrical shells.
• Frustum: A portion of a cone formed by cutting off the top with a plane parallel to the base.

Timeline

• Mid-semester: Mid-semester test 3 results: Complex numbers identified as difficult.
• Week 13: Focus on 'Week 13' content on Moodle for exam preparation, including mock exams.
• June 1 - June 18: Pre-exam consultation sessions held by Jessy and the instructor.
• Until June 19: Maths Learning Centre open until the second week of the exam period.
• Week 12: Applied classes and online assignments cover material from Week 11 and Week 12.

People

• Daniel Mathews: Instructor/Lecturer
• Jessy: Teaching Assistant/Support Staff