Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

13: Integrals

( \newcommand{\kernel}{\mathrm{null}\,}\)

”Success is the sum of details” - Harvey Firestone

In a seemingly unrelated turn, we shift our focus to a new question: how to find the area under a curve. Chapter thirteen explores exactly how we can use limits to find exact areas. We start by refreshing ourselves on our area formulas from geometry, and soon run into limitations for some functions. We start by trying to estimate the area under the curve by using rectangles, and soon after learn that if we incorporate a limit we have defined a new object: the definite integral. This tool (much like the limit definition of derivative) is a cumbersome tool for finding the area under a curve.

Just like for derivatives, we quickly look for patterns to help avoid doing as much work. We discuss the properties of integrals, and how they are strangely similar to some of the properties of derivatives. This is no coincidence, as we discover the Fundamental Theorem of Calculus: derivatives and integrals are opposites. We hijack the integral sign to create the indefinite integral which stands for anti-derivatives in general. After a little bit of practice, we realize that indeed many of our differentiation skills can double as anti-derivative skills.

Lastly we turn back to the Fundamental Theorem of Calculus and how it applies to us solving definite integrals. We can see the wide variety of different functions we can integrate, and even some real world applications to suggest why we would like to do so. Just be cautious that there are many more tricks we will need before we are professional anti-derivative takers.


This page titled 13: Integrals is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Ken Huber.

Support Center

How can we help?