Week 2 Notes
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This 2 page Class Notes was uploaded by Bethany Lawler on Friday September 4, 2015. The Class Notes belongs to Math 182 at Washington State University taught by S. Lapin in Summer 2015. Since its upload, it has received 70 views. For similar materials see Honors Calculus II in Math at Washington State University.
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Date Created: 09/04/15
Math 182 Notes Week 2 831 94 Volume by Slicing o The volume of any three dimensional shape can be found by taking a cross section of the solid 0 Once the area of the cross section Ax is found the area can be found by the following formula b Ax dx a Disk Method 0 The volume of a solid created by rotating a function around an axis can be determined using disk method 0 It is assumed that the cross sections of the shapes are disks with a radius equal to the function value at any given point 0 Therefore the following formula based on TCT39Z can be used to determine the volume b 7Ifx2 dx Washer Method 0 The volume of a solid created by rotating the area between two functions around an axis can be determined by the washer method 0 It is assumed that the cross section of this shape is a washer or a disk with a portion of the center missing 0 It therefore makes sense to treat the overall shape as though it could be cut into disks and subtract the smaller inner disks from the larger out disk 0 The formula for finding volume via the washer method is b j nfx2 gcx2dx Washer and Disk Method with respect to Y 0 When using the washer or disk method with respect to y solve all equations for x c Find the limits points of intersection in terms of y o The disk equation in terms of y is d moo2 dy o The washer equation in terms of y is d j new qcy2 dy Rotation around a line other than an axis 0 When rotating around a line other than an axis the radius of disks and washers must be adjusted To adjust the radius for a function below or to the left of the line of rotation subtract the function value from the value of the line of rotation To adjust the radius for a function that is above or to the right of the line of rotation add the function value to the value of the line of rotation Use the adjusted radius values in the washer or disk equations to find the volume Shell Method Shell method is used to slice the shape into cylinders perpendicular to the line of rotation Cylinders of the shape created by rotating a function around the y axis have a circumference of 27tx a height of fx and a width of AX The formula for shell method is b 271x fx dx To use shell method to find the volume of a shape created by the area between two functions use the following formula lb2nx fx dx gx dx Using shell method with holes in the shape To use shell method when there are gaps in the shape change the limits of the integral If there is a hole from X 0 to X r the integral changes to frb 271x f x dx Length of curves The length of a curve can be calculated using the following integral b j 1 F x2 dx
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