Introduction

Welcome to the FiveHive article for Unit 6.6 of AP Calculus! 

In this article, we will be looking at what everyone has been waiting for since unit 5: solving integrals!

As usual, we will only cover the topics included in the CED for unit 6.6.

Properties of Definite Integrals

To start off our exciting journey into integration, let’s look at some integration rules. These will be extremely important in solving all types of integrals, so it is imperative that you understand all of these. 

  NOTE: For the scope of this article, we will be referring to the numbers on the top and bottom of the integral sign as limits or bounds interchangeably. 

Equal Bounds: If the integral is calculated with a lower bound which is equal to the upper bound, the value of that integral will be 0.

${\int }_g^{g}f(x)dx=0$

Reverse Bounds: A rule in integration is that you can only integrate when the lower bound is less than the upper bound. In the case that the bounds need to be flipped, you can simply multiply the integral by negative 1.

${\int }_3^{-2}f(x)dx=-{\int }_{-2}^{3}f(x)dx$

Integration with a constant multiplied: If the function being integrated is multiplied by a constant, you can take the constant to the outside and integrate from there. 

${\int }_a^{b}kf(x)dx=k{\int }_a^{b}f(x)dx$ where k is a constant

Addition of Integrals: If two integrals of a certain function are being added whose bounds are adjacent, you can convert that expression into one integral.

${\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx={\int }_a^{c}f(x)dx$ where $a<b<c$ 

Splitting an Integral: If you are integrating two functions being added or subtracted on the same bounds, you can split that expression into two integrals.

${\int }_a^b[f(x)\pm g(x)]dx={\int }_a^{b}f(x)dx\pm {\int }_a^{b}g(x)dx$

Symmetry: If you have an odd function being integrated from a negative number to a positive number of the same value, the value will be 0. For even functions, the value will be two times the integral of the function from 0 to the positive bound.

Odd function: ${\int }_{-a}^{a}f(x)dx=0$

Even function: ${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx$

Practice

WHOOO! You have just made it through the first part of learning how to solve integrals! How interesting?!

Now, it’s time to put what you have learned to the test with some practice questions.