Shell Method
int(a,b) (2pirh)
The Shell Method is a technique used in calculus to calculate the volume of a solid of revolution, which is when a two-dimensional shape is rotated around a specific axis to form a three-dimensional object.
To use the Shell Method, the first step is to identify the axis of rotation and choose a representative vertical line segment parallel to the axis. Next, the segment is rotated around the axis to form a cylindrical shell with a height equal to the length of the segment, and a radius equal to the distance from the axis to the segment.
The formula for calculating the volume of the shell is:
V = 2πrhΔx
where V is the volume, r is the radius, h is the height, and Δx is the width of the shell.
To find the total volume of the solid, we integrate the volume of each shell over the interval of rotation. The integral can be set up as:
V = ∫ 2πrh dx
where r and h are functions of x.
The Shell Method is often used when the cross-sectional area of the solid is difficult to find or not constant, and the axis of rotation is vertical. It can also be used for solids of revolution with horizontal axes of rotation, but in that case, a representative horizontal line segment would be chosen instead of a vertical one.
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