1. By: Rossboss, Chase-face, and Danny “the Rock” Rodriguez

2. Formulas Area of a region between two curves Volume by Disk Method Volume by Washer Method Volume of Solids with Known Cross Sections Volume by Shell Method The area formula can be modified to fit an equation in terms of y by inserting y in the positions of every x All of the given volume formulas are for rotation around the x-axis. For rotation around the y-axis, you must put the equation in terms of x to y or y to x.

3. Strategies and Hints For finding area between two curve and certain circumstances for finding volume, if the function is evenly split by an axis you can only integrate half of it then multiply your answer by 2. To figure out the bounds of the function you must set the functions equal to each other and solve for x where they intersect. It is helpful to graph the functions on your calculator to both find the intersecting points for you bounds as well as seeing the graph so you know what you will know which parts you will have to subtract from each other. On problems with terms that you are unable to integrate (ln for example) you can enter the integral in your calculator to give you an estimated answer. You can also use this function to check your answers. Using your calculator: MATH > 9:fnInt( > function, define your term, lower bound, upper bound ) If the equations given for you to solve for a volume is being rotated around a horizontal axis of revolution, you would use an equation in terms of x. If it is being rotated around a vertical axis you would use an equation in terms of y. (This applies to the disk and washer method)

4. Area Between Two Curves When calculating the area between curves it is important to have the correct set up. You need to combine the two function and integrate them. To do this, in terms of x, you take the upper function and subtract the lower function, or in terms of y take the rightmost and subtract the left most.

5. Volume by Disk Method You would use the disk method to find the volume of area rotated around an axis or specific line line if the section shares a complete side with the axis of rotation. The cross sections of the solid will create disks.

6. Volume by the Washer Method You would use the washer method, which is similar to the disk method in process, when there the solid does not share a complete side with the line of rotation. This space created when it is rotated makes the cross section of these solids circular but with holes in the middle of them. (washers)

7. Volume of Solids with Known Cross Sections You would use this method when you have a known cross section and an equation that bounds its base. Common cross-sections are squares, rectangles, triangles, semicircles, and trapezoids.

8. Volume by the Shell Method The washer method can be interchanged in some situations with the shell method. It can sometimes be more convenient if the equations are in terms that doesn’t agree with the axis of rotation. With this method you can rotate solids made by equations in terms of x on a vertical axis and vise versa.

9. Helpful Websites http://www.mecca.org/~halfacre/MATH/appint.htm http://www.wyzant.com/Help/Math/Calculus/Integration/Finding_Volume.aspx http://curvebank.calstatela.edu/volrev/volrev.htm http://www.msstate.edu/dept/abelc/math/volume.html