The Shell Method Volume of Revolution
Use the shell method to determine the volume formed by the bounded region rotated about the x-axis.
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The shell method is another approach for finding the volume of a solid created by rotating a curve around an axis. Instead of slicing the solid into disks, the shell method involves cutting the solid into thin, hollow cylindrical shells, which are then stacked to form the entire volume.
In this problem, we are revolving the function around the y-axis, which means that as we rotate the function, we create cylindrical shells. Each shell represents a small section of the solid. The height of each shell corresponds to the value of the function, while the radius of each shell is the distance from the y-axis (the axis of rotation) to the function. The thickness of each shell comes from how much the function changes over a small interval.
Conceptually, the shell method works by calculating the volume of each thin cylindrical shell, where the surface area of the shell is multiplied by its thickness. These shells are then "peeled" off and stacked together to find the total volume. The key difference from the disk method is that the shell method is often more convenient when rotating around a vertical axis, especially when the function is expressed in terms of x, as it avoids solving for x in terms of y.
In this specific problem, as we revolve the function around the y-axis, we imagine hollow cylinders formed by the curve y equals x squared. By summing up the volumes of these shells across the entire interval, we calculate the total volume of the solid. The shell method is particularly useful for problems like this because it simplifies the integration and calculation process when dealing with rotation around a vertical axis.