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 wiggle 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 between 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!

**Objectives**

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

- 4.2.1 Sketch rectangles based on left approximation, right approximation, or midpoint approximation
- 4.2.2 Understand if the left or right approximation will be more accurate based on the shape of the curve
- 4.2.3 Set-up and evaluate (by hand) an integral that computes a Riemann Sum
- 4.2.4 Understand that a definite integral is a limit of a sum of the areas in an infinite number of rectangles
- 4.2.5 Evaluate a definite integral of a basic function (by hand)
- 4.2.6 Use a definite integral to find the exact area under a non-negative curve

**Terminology**

Define: Riemann sum, left/right/midpoint approximations, area under a curve, definite integral, limits of integration

**Supplemental Resources (recommended)**

Explore Riemann sums with the Riemann Sums Applet from Hobart and Williams Smith Colleges.

**Supplemental Resources (optional)**

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.

Lesson: Sigma Notation and Riemann Sums, Dale Hoffman's Contemporary Calculus

Lesson: The Definite Integral, Dale Hoffman's Contemporary Calculus

Lesson: Properties of the Definite Integral, Dale Hoffman's Contemporary Calculus

Lesson: Areas, Integrals, and Antiderivatives, Dale Hoffman's Contemporary Calculus