Let's say you are given the equation y = x 2/3įind the volume of the solid when this function is rotated about the x axis in the domain In this case, to find the volume of the rotated sphere, we would instead use cylindrical shells. Usually, this is because for part of the graph, the radius would be calculated by subtracting one part of a curve from itself. However it is important to note that under some circumstances, plane slicing does not work and thus should not be used. Also, in order for this technique to work, the limits of integration, integrand, and differential all must be in terms of the same variable. It is important to note that the plane slice is always perpendicular to the axis of rotation. To find the volume of the entire solid, we add up the volumes of the disks formed from plane slices at each point x in the interval. The volume of a disk is the area of the base (πr 2) times the width (dx) therefore the volume of this disk (dV) is equal to πr 2dx. We can then substitute "y" of each equation in terms of "x." So, in the example above, the radius would be equal to x 1/3 - 0, or x 1/3.
The radius is equal to the distance from the curve and the axis of rotation (x-axis) or Y curve - Y axis. The plane "slice" becomes the shape of a disk. In Figure 2, this graph is rotated about the x-axis and a circular plane of thickness dx is "sliced" through the point (x,y).įigure 2 shows the solid formed by rotating the equation in Figure 1 around the x-axis.
The graph of y=x 1/3 is shown below in Figure 1. This axis is most often the x or y-axis, although it can be different. Plane slicing is used to find the volumes of solids that are generated from rotating an equation or equations, bound by a specific domain, around a specified axis. In order to find the volume of such solids, you must use calculus integration techniques, such as plane slicing. For irregular objects, the dimensions for the area of the base are not uniform throughout the figure. Volume of a Solid of Revolution by Plane Slicingįor regular objects, the volume of a solid is the area of its base times its length.
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