Parabolic Integration

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{{Image Description |ImageName=Real Life Parabolas |Image=Golden Gate Bridge.jpg |ImageIntro=Parabolas are very well-known and are seen frequently in the field of mathematics. Their applications are varied and are apparent in our every day lives. For example, the main image on the right is of the Golden Gate Bridge in San Francisco, California. It has main suspension cables in the shape of a parabola. Of course there are many more examples of parabolic architecture such as roller coasters, flight paths, and probably the most recognized, the Golden Arches of McDonald's. With all of these appearances in real life, have you ever wondered how to find the area under one? |ImageDescElem=Two methods for finding parabolic area exist. One is very accurate and the other is more of an approximation method. This procedure for approximation is known as the Rectangle Method and is used by finding the area of rectangles that can fit in the parabola. The area of these rectangles are added together, giving you the approximate area under or above the parabola.

For a detailed overview of parabolas, see the page, Parabola. However, we will provide a brief summary and description of parabolas below before explaining how to find the area beneath or above one.

Basic Definition

You may recall first learning about parabolas and your teacher telling you that it is a curve in the shape of a "u" and can be oriented to open upwards, downwards, sideways, or diagonally. To be a little more mathematical, a parabola is a conic section formed by the intersection of a cone and a plane. Below is an image illustrating this.

Image:Conic Section Parabola.jpeg

When you were first introduced to parabolas, you learned that the quadratic equation,  y= a(x-h)^2+ k is its algebraic representation (where h and k are the coordinates of the vertex and x and y are the coordinates of an arbitrary point on the parabola.

As you progressed in mathematics, you learned how to find the area of the space enclosed by the parabola. This can be accomplished in two different ways:

  • Using Definite Integration
  • Using the Rectangle method (Also referred to as finding the Riemann Sum)


We will explain both of these approaches by posing a problem and then solving it step by step. But first we are going to familiarize you with some parabolic architecture and occurrences found in the real world.


|ImageDesc===Integral Approach to Determining Area== Typically, when attempting to find the area underneath a parabola, we take its integral. Below is a proposed problem with a numbered procedure of individual steps for completion:


Find the area under the curve y =4-x^2 between x = 0 and x = 2 and the x-axis.


1.Graphing the function first will help you to visualize the curve. Below I have graphed the function using the mathematical program Derive, but you can easily graph it either using your calculator or by hand.

Image:Derive-Parabola.jpeg


2.Now we will algebraically evaluate the expression by taking its integral; doing so will give us an EXACT area. Integration is shown and calculated by:


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