In the previous module you defined left and right Riemann sums and used these sums to find the area under the curve f(x) = x² from x = 0 to x = 1. In this module you will use Riemann sums to find other areas and discover the remarkable connection between areas under a curve and derivatives called the Fundamental Theorem of Calculus. Although you will discover this theorem in the context of finding areas under a curve, it is used in a wide variety of applications extending far beyond finding areas.

Lesson Index:

16.1 - Area Functions, A Symbolic Approach

16.2 - Area Functions, A Visual Approach

16.3 - The Fundamental Theorem of Calculus

After completing this module, you should be able to do the following:

• Describe the Fundamental Theorem of Calculus using a symbolic approach
• Recognize the Fundamental Theorem of Calculus in a graphical setting
• Use the Fundamental Theorem of Calculus to evaluate definite integrals and find areas
• Visualize the general shape of an area function by using characteristics of the curve function
• Use the Fundamental Theorem of Calculus to differentiate functions defined by integrals