Study Guide

Definite Integrals

Definite Integrals Introduction

It's time to answer the burning question that every calculus student asks. How does all of this "calculus" stuff relate to geometry? No? That question hasn't come to mind? Maybe we're trying to plant thoughts in your head, Big Brother style. Either way, calculus does relates to geometry. This section will be dedicated to areas. Area under the curve, area of a triangle, Area 51, definite integrals can be used to solve them all; maybe not the alien thing.

If you aren't convinced about the super usefulness of definite integrals—drumroll— they can also be used to calculate volumes. That's right. You want to know how much water fits in your grandma's vase? Integrate that vase. How much cheese is actually in the wheel? Integrate that cheese.

In this unit we'll show you tons of ways to approximate funny-shaped areas. Some areas we'll be able to find exactly. We'll tell you what integrals are and useful things you can do with them.

Definite Integrals Resources

Wait, what happened to derivatives? What are these squiggly S-looking symbols? Here's a brief description of the mysterious invaders.

Approximating Definite Integrals with Midpoint and Trapezoidal Sums:
Back to sketching things, this time we are drawing rectangles and trapezoids beneath curves. Feel like a 5th-grader again, and learn more about definite integrals.

Properties of Definite Integrals With Examples:
Integrals feel pretty complicated sometimes, so here's a list of properties you can use to simplify them a little. We won't take off your shoes and give you a pedicure, but we'll at least tell you our favorite places to go.

Integrating Even and Odd Functions:
Trick or treat? Have both. You have an integral with an odd or even function for an integrand. Before you go trying to solve it, use one of these tricks.


Estimating the Area Under a Curve:
Calculus is older than sliced bread. It makes you wonder how mathematicians explained areas under curves before it. But since we have sliced bread, make a sandwich and watch this video.

The Definite Integral – Understanding the Definition:
If you're still having trouble wrapping your brain around integrals, this video will make your brain a flour tortilla and integrals the chicken, cheese, salsa, sour cream, and guacamole filling.

How to Solve a Basic Definite Integral:
A simple example, explained. It's so clear, you'll swear we cleaned the computer screen with Windex before playing the video for you.

Properties of Definite Integrals:
Having trouble understanding how two integrals become one? Understand how these mathematical amoebas merge with explanations and examples.

Games and Tools

Riemann Sum Calculator:
It's tedious drawing 1000 little rectangles under a curve. We've employed a graphics artist to do it for you.

Definite Integral Calculator:
Not all definite integral calculators are equal. This is one of the best. Check the answers to your homework here, and use the extra free time to play some Super Mario.