Integrals

Prerequisite skills: Must be familiar with deriving an equation using the power and chain rules Must know derivatives of trig. functions  Must be familiar with proper exponent notation
 * __ SUMMARY OF INTEGRALS  : Bailey Mann __**

What is an Integral?: An integral in the notation used for the anti-derivative of a function that can also be used to represent the equation for the area under a curve. Integrals can be split into two sub categories: indefinite integrals and definite integrals. When solving an indefinite integral you must include the constant of integration and will have a function with two variables as an answer. If the problem provides an initial condition for an indefinite integral then you can solve for the constant of integration and find the original function with only one variable. When solving a definite integral the upper and lower bounds provided will result in an answer that is a constant.

Steps to solving an integral: Indefinite integrals 1. Raise the power of every term in the integrand by one then divide by that power - u substitution can be used if the function you are integrating includes a function and its derivative - to use u substitution rewrite the integral with u representing the function and du representing its derivative - if only a portion of du is provided add a coefficient that accounts for that then integrate normally - after integrating substitute u back in 2. Simplify any coefficients 3. Add your constant of integration to the end of the equation

Definite Integrals 1. See step 1 of indefinite integrals <span style="background-color: #ffffff; color: #222222; font-family: Calibri,sans-serif; font-size: 15px;">2. Evaluate the equation at the upper limit then subtract the equation evaluated at the lower limit <span style="background-color: #ffffff; color: #222222; font-family: Calibri,sans-serif; font-size: 15px;">3. Simplify

<span style="background-color: #ffffff; color: #222222; font-family: Calibri,sans-serif; font-size: 15px;">Area Under the Curve <span style="background-color: #ffffff; color: #222222; font-family: Calibri,sans-serif; font-size: 15px;">1. Find the equation of the curve <span style="background-color: #ffffff; color: #222222; font-family: Calibri,sans-serif; font-size: 15px;">2. Integrate curve equation <span style="background-color: #ffffff; color: #222222; font-family: Calibri,sans-serif; font-size: 15px;">3. Treat as definite integral is limits are given <span style="background-color: #ffffff; color: #222222; font-family: Calibri,sans-serif; font-size: 15px;">4. If limits are not given use limit notation to find the approximation as the variable of derivation approaches infinity

<span style="background-color: #ffffff; color: #222222; font-family: Calibri,sans-serif; font-size: 15px;">Formulas, Theorems, Rules:

<span style="background-color: #ffffff; color: #222222; font-family: Calibri,sans-serif; font-size: 15px;">Common Integrals \displaystyle \int udx \pm \displaystyle \int vdx \pm \displaystyle \int wdx \pm \cdots $"]] __** \int f(u)du$"]] __** \frac{dx}{du}du = \displaystyle \int \displaystyle \frac{F(u)}{f'(x)}du$"]] __**
 * __ 1. __**[[image:http://www.sosmath.com/tables/integral/integ1/img1.gif width="100" height="53" align="MIDDLE" caption="$\displaystyle \int adx=ax$"]]
 * __ 2.[[image:http://www.sosmath.com/tables/integral/integ1/img2.gif width="205" height="53" align="MIDDLE" caption="$\displaystyle \int af(x)dx=a \displaystyle \int f(x)dx$"]] __**
 * __ 3.[[image:http://www.sosmath.com/tables/integral/integ1/img3.gif width="459" height="53" align="MIDDLE" caption="$\displaystyle \int \left( u \pm v \pm w \pm \cdots \right) dx =
 * __ 4.[[image:http://www.sosmath.com/tables/integral/integ1/img4.gif width="172" height="53" align="MIDDLE" caption="$\displaystyle \int udv = uv - \displaystyle \int vdu$"]] __**
 * __ 5.[[image:http://www.sosmath.com/tables/integral/integ1/img5.gif width="208" height="58" align="MIDDLE" caption="$\displaystyle \int f(ax)dx = \displaystyle \frac{1}{a} \displaystyle
 * __ 6.[[image:http://www.sosmath.com/tables/integral/integ1/img6.gif width="352" height="63" align="MIDDLE" caption="$\displaystyle \int F\{f(x)\}dx = \displaystyle \int F(u) \displaystyle


 * __ 7.[[image:http://www.sosmath.com/tables/integral/integ1/img7.gif width="205" height="64" align="MIDDLE" caption="$\displaystyle \int u^{n}du = \displaystyle \frac{u^{n+1}}{n+1}, n \neq -1$"]] __**



-Caylin McCubbin

__**  PRACTICE PROBLEMS  : **__ Evaluate the definite integral of the algebraic and trigonometric function. To check your work double-click on the problem.


 * **Beginner:**

|| **Getting There:**

||
 * **On the Way:**




 * Rock Star :**

(try U- substitution ) ||



||
 * For more help check the links below. -Michael White ||
 * __  INSTRUCTIONAL VIDEOS  : __**

Introduction to Anti-derivatives and Indefinite Integrals

Indefinite Integrals of "x" Raised to a Power

Anti-derivatives of SCARY Looking Expressions

Anti-derivatives With "x" Raised to a Negative Power (want to take a look ahead? this video discusses natural logs briefly)

Anti-derivatives of Basic Trig Functions/ Exponential Anti-derivatives

Introduction to U-Substitution

More U-Substitution

And Some More U-Substitution

U-Substitution of Definite Integrals

Simple Riemann Approximation (using rectangles)

Left Riemann Sums

Rectangular and Trapezoidal Riemann Approximation

Trapezoidal Approximation of Area Under a Curve

Riemann Sums and Integrals

-Caylin McCubbin

__** HELPFUL SITES : **__ Mernuelita Florissant

<span class="wiki_link_ext">Overview and examples of the [| Fundamental Theorem of calculus] <span class="wiki_link_ext">Techniques of Integration __[|: Substitution]__ <span class="wiki_link_ext">A summary of definite integrals with examples __[|: Definite Integrals]__ <span class="wiki_link_ext">Overview of __[| Indefinite Integral]__ __[|Integrals of Trigonometric Function]__

**REAL-LIFE APPLICATION:** Mernuelita Florissant 1. An electrical engineer uses integration to determine the exact length of power cable needed to connect two substations that are miles apart. 2. An architect will use integration to determine the amount of materials necessary to construct a curve domed over a new sports arena, as well as calculate the weight of that dome and determine the type of support structure required.

__** ANNOTATED BIBLIOGRAPHY: Bailey Mann **__ Hockett, Shirley O. //Barron's AP Calculus//. 10th. Hauppaug, NY: Barron's Educational Series, Inc., 2008. 215-290. Print.

The Barron's 10th Edition of the AP Calculus review book splits antidifferentiation and definite integrals into two seperates chapters. Both chapters 5 and 6 break down the material, provide example problems with solutions, and provide helpful formulas, tiops, and tricks for the test. The book contains material for BC test prep that does not apply to our course, but highlights theBC materials throughout the chapter. If when using this review book, the student comes across material that seems to not apply to lessons in class, make sure to double check the left margain for the BC notation.