Ring Method and Shell Method
The first-quadrant area is bounded by the curve , the x axis, and the line x = 4 is rotated about the y axis. Find the volume generated: (a) By the ring method (b) By the shell method
In this problem, we are dealing with the area bounded by the curve, the x-axis, and a vertical line, and we are rotating this region about the y-axis. We’ll calculate the volume of the solid created by this rotation using two different methods: the ring method and the shell method.
(a) Ring Method:
The ring (or disk/washer) method involves slicing the solid perpendicular to the axis of rotation (in this case, the y-axis). Each slice or cross-section is a circular ring or disk, depending on whether the inner radius is non-zero. The idea is to stack these rings to build up the total volume of the solid.
In this problem, we are rotating around the y-axis, so each ring corresponds to a horizontal slice of the region. The outer radius of each ring is the distance from the y-axis to the curve, and the inner radius would be zero since there’s no hole in the middle (it's a solid disk). To use the ring method, we express the curve in terms of y, so that the radius of each disk depends on y. We then calculate the volume by adding up the areas of these disks multiplied by their thickness (a small change in y). The total volume is obtained by integrating these disks over the range of y values that the region occupies.
(b) Shell Method:
The shell method, on the other hand, involves slicing the solid parallel to the axis of rotation. Instead of stacking disks, we now think about peeling off thin cylindrical shells and stacking them together to form the solid. For each shell, its height corresponds to the height of the curve, its radius is the distance from the y-axis (the axis of rotation), and its thickness is a small change in x.
In this case, we use the shell method by calculating the volume of each cylindrical shell. The radius of each shell is simply the x-value at a given point, and the height of the shell is the y-value (or how high the curve reaches at that x-value). The volume of each shell is found by multiplying the surface area of the cylindrical shell by its thickness. By summing up the volumes of these shells over the interval from x equals zero to x equals four, we can find the total volume.
In summary, both methods compute the volume by slicing the solid in different ways—either perpendicular (ring method) or parallel (shell method) to the axis of rotation. The choice of method often depends on how the function is expressed and which method leads to simpler integration for the specific problem at hand.