Applications+of+Integrals

Unit 6: Applications of Integrals -Marissa Powell =**__Notes- Triston Payne__**= > -Area is, by definition, a positive quantity. In order to find the area between two curves, you need to determine on which interval(s) one curve is above the other, so that the difference of the two functions is positive. - Examples: Triston Payne- Site - The difference between the Area Between Two Curves and Intersecting Curves is that Two Curves don't intersect and points a and b are given. Area of Intersecting, points a and b must be found but setting the two equations equal to each other and solve for x. -Examples: Triston Payne- Slideshow(1-4) A solid of revolution is formed by rotating a two-dimensional function around an axis to produce a three-dimensional shape. The volume //V// of a solid of revolution is or , depending on the axis of rotation. The cross-sectional area of the solid is, where //r//0 is the outside radius and //ri// is the inside radius of the solid (in the case of a full solid,  ), dependent on the function and the axis of rotation. -Examples: []Triston Payne- Site - The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a washer.To obtain a "hollow" solid of revolution, the procedure would be to take the volume of the inner solid of revolution and subtract it from the volume of the outer solid of revolution. About the //x//-axis: ; where //f//(//x//) = outer radius &//g//(//x//) = inner radius. -Examples: Triston Payne- Site
 * **Area of a Region Between Two Curves**-If f and g  are continuous on [ a , b ]  and g ( x )≤ f ( x )  for all x  in <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">[ <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">a <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">, <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">b <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">] , then the area of the region bounded by the graphs of <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">f  and <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">g  and the vertical lines <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">x <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">= <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">a  and <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">x <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">= <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">b  is <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">A <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">= <span class="mo" style="font-family: MathJax_Size1-Web; font-size: 17.73px; vertical-align: 0px;">∫ ^ <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 12.4px;">b a <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">[ <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">f <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">( <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">x <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">)− <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">g <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">( <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">x <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17.73px;">)] <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17.73px;">dx
 * **Area of a Region Between Intersecting Curves**
 * **The Disk Method**
 * -** Another application of Integrals is finding the volume of a three-dimensional solid. The disk method is a way to calculate the volume of a solid of revolution by taking the sum of cross-sectional areas of infinitesimal thickness of the solid.
 * **The Washer Method**


 * __Videos__**
 * __[]__**
 * This video is very helpful in understanding the rotating of a function about the x-axis. It breaks down step by step the process of solving for the volume of the rotated figure. Also it gives a very useful visual representation of what is happening when the figure is being rotated. --Marissa Powell**

[|Troy Askew- Video]. Finding Volume the disk and washer methods. [|Troy Askew- Video] D&W method, x-axis,y-axis,2 intersecting functions. [|Troy Askew- Video]. Shell Method [|Troy Askew-Video]. Longer Version of disk very deep explanation.

__**Example Problems-Marissa Powell**__

[]- This site allows you to download different examples of applications of integrals on the "Nspire" calculator and also shows you how to use your calculator to help you find the answer.-**Josh Stowe** []- This useful site reviews definite integrals and explains them in a simple manner and then introduces you to applying them and gives helpful tips on how to remember what steps to take.-**Josh Stowe** []- This site can be especially helpful to people who have been struggling to find volume with definite integrals, as it specifically deals with that part of applying integrals.-**Josh Stowe**
 * Sites:**
 * [|Troy Askew-Site]**
 * This site is very helpful has pictures to examples with equations showing how to solve and visualize the graphs.**
 * []**- This website has different links that give you options to work on each part of applying integrals. It also gives you practice problems and notes.-**Josh Stowe**

You can apply definite integrals to a situation with two cars going down a highway at two different speeds/velocities over the same period of time. To find the distance between the two, you would have to use applications of a definite integral. Also, applying integrals when dealing with area and volume can be used to measure items such as buildings downtown in a large city in order to see how they will fit in with other buildings and skyscrapers around them while maintaining even symmetry.
 * __Real-Life Issues__** -**Josh Stowe**