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JKLM
- 21
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I don't understand how to set up the washer and shell equations. When you are given the function and the line to rotate it around, or two functions and a line.
The washer method and the shell method are both techniques used to calculate the volume of a solid of revolution. The difference between the two lies in the shape of the cross-section used to create the solid. The washer method uses circular cross-sections, while the shell method uses cylindrical shells.
The choice between using the washer method or the shell method depends on the shape of the solid and the axis of revolution. If the solid has a circular base and the axis of revolution is perpendicular to the base, the washer method is usually used. If the solid has a non-circular base or the axis of revolution is parallel to the base, the shell method is typically used.
The formula for calculating volume using the washer method is V = π∫a2-b2dx, where a and b represent the outer and inner radii of the cross-section, and dx represents the width of the cross-section. This integral is typically evaluated over the interval of integration that corresponds to the axis of revolution.
The formula for calculating volume using the shell method is V = 2π∫r(x)h(x)dx, where r(x) represents the distance from the axis of revolution to the outer edge of the shell and h(x) represents the height of the shell. This integral is typically evaluated over the interval of integration that corresponds to the axis of revolution.
Yes, both the washer method and the shell method can be used to calculate the volume of solids with holes. In these cases, the formula for calculating volume may need to be adjusted to account for the hole or holes in the solid. This can be done by subtracting the volume of the hole(s) from the total volume calculated using the respective method.