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


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 :)


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


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

STUDY: Riemann Sums, the Definite Integral, and Area Under the Curve

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.

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

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

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

rev. 2020-11-16