Timeline 2013 2009 2010 2011 2012 0 + - = 9 8 7 1 2 3 4 5 6 c Sample Problem 3. Find the volume of the solid generated by revolving the region bounded by the y-axis and the curve for 1<y<4 about the x-axis using the shell method. Sample Problem 2. Find the volume of the solid of revolution obtained by rotating the region in the first quadrant bounded by

about the y-axis. Problem Collection 1. Find the volume of the solid of revolution obtained by rotating the region under the curve of

about the x-axis on the interval 0<x<1. Anik, Crystal,

Esha, Jonathan HERE WE GO!!! Disk Washer VOLUME: DISK, WASHER, AND CYLlNDRICAL SHELLS The Disk Method is a means of calculating the volume of a solid of revolution.

If the function to be revolved is a function of x, the following integral represents the volume of the solid of revolution:

R(x) is the distance between the function and the axis of rotation. Axis of rotation is horizontal.

If the function to be revolved is a function of y, the following integral will obtain the volume of the solid of revolution:

R(y) is the distance between the function and the axis of rotation. Axis of rotation is vertical. PAY ATTENTION OR YOU WILL GET A ZERO BIGGER THAN YOUR HEAD!!!!! Shells Shell integration (cylindrical shell method) is a means of calculating the volume of a solid of revolution when integrating along an axis parallel to the axis of revolution. In this method a part of the graph of a function is rotated around an axis, and is modeled by an infinite number of hollow "strips" all infinitely thin. To calculate the volume when the rotation is around the x-axis the equation is such:

However if the rotation is around the y-axis (vertical axis of revolution) the equation is as follows: 0 Sample 4. Find the volume of the solid obtained by rotating the region below the graph of

about the x-axis for

. Sample Problem Esha Crystal Anik Jon Mrs. Cisnero The disc method can be extended to cover solids of revolution with a hole by replacing the representative disc with a representative washer. The washer is formed by revolving a rectangle about an axis.

To find the volume of a solid of revolution created by a washer, use the formula below.

Note that the integral involving the inner radius represents the volume of the hole and is subtracted from the integral involving the outer radius. Problems Summary EXPLANATION FOR SAMPLE 1 DISK WHEN TO USE EACH METHOD one must draw a cross-section from the curve to the axis of rotation if the line is perpendicular to the axis of rotation washer method disk method if the line is parallel to the axis of rotation shell method SHELLLLLLLLL EXPLANATION TO SAMPLE 2 EXPLANATION TO SAMPLE 3 EXPLANATION TO SAMPLE 4 you are very esmart ;) IF THE CROSS-SECTION TOUCHES THE AXIS OF ROTATION DISK if the cross-section does not touch the axis of rotation perpendicular to y-axis perpendicular to x-axis use disk with differential dx use disk with differential dy

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