Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Calculus I
Topic: Antiderivatives
Subtopic: Riemann Sums, Definite Integrals, and Area Under the Curve

Overview

Calculus can be used to find the area of non-standard shapes such as the area between a wiggly curve and the x-axis. Breaking this region into lots of thin rectangular strips and adding their area approximates the total area under the curve. Using more rectangles, thinner ones, improves the approximation. This is the idea behind Riemann Sums. Letting the number of rectangles approach infinity and taking the limit of the total area will give the exact area under the curve. This process is accomplished using a definite integral which is an integral with specific starting and ending x-values. Lots of visual concepts and algebraic manipulation to pull together in this lesson!

Please keep the "big picture" of Calculus in mind as we see this again and again: Breaking things into smaller and smaller pieces, but more and more of them, then taking a sum (integral) of all the tinier and tinier pieces as the number of pieces goes to infinity (limit). This gets us to the entire quantity (eg. area). That's rough but the main idea :)

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 4.2.1 Geometrically interpret integration as finding the area between two curves
• 4.2.2 Sketch rectangles based on left approximation, right approximation, or midpoint approximation
• 4.2.3 Understand which of the left or right approximation will be more accurate based on the shape of the curve
• 4.2.4 Correctly use and "read" sigma/summation notation
• 4.2.5 Set-up and evaluate (by hand) an integral that computes a Riemann Sum
• 4.2.6 Understand that a definite integral is a limit of a sum of the areas in an infinite number of rectangles
• 4.2.7 Evaluate a definite integral of a basic function (by hand)
• 4.2.8 Use a definite integral to find the exact area under a non-negative curve

Terminology

Terms you should be able to define: sigma, summation, Riemann sum, left/right/midpoint approximations, area under a curve, definite integral, limits of integration

Mini-Lectures and Examples

Supplemental Resources (recommended)

An applet that introduces integration well worth exploring: Gaining Geometric Intuition

Supplemental Resources (optional)

Two more applets that introduce integration that you may enjoy exploring:
Bicycle Problem (part 1) and Bicycle Problem (part 2)

Video: The Area Under a Curve, Selwyn Hollis's Video Calculus

Video: The Integral, Selwyn Hollis's Video Calculus includes signed area and a geometric interpretation of the integral.

rev. 2020-11-16