Stroll Calculus 34-volumetric calculation: cylindrical Shell method

There is also a way to go volume, often it is more convenient than the method of the previous article.

To understand this approach, consider the area shown on the left side of Figure 1, that is, the first quadrant axis and the area shown by the curve y=f (x) y=f (x). If the area rotates around the x x axis, then the vertical narrow band in the figure generates a disk, and we can get the total volume of the disks from the X=0 x=0 to the x=b x=b interval. This is, of course, the disc method described in the previous article. However, if the area rotates around the y y axis, as in the middle of the picture, then we get a completely different object, and the vertical narrowband produces a thin cylindrical shell. The shell can be seen as a tin, but the top and bottom have been removed, or very thin cardboard. Its volume DV DV is essentially the inner cylindrical surface area (2πxy) (2\pi xy) multiplied by the thickness (dx) (DX), so
DV=2ΠXYDX (1) \begin{equation} dv=2\pi xydx\tag1 \end{equation}

The radius x x of this shell is increased from x=0 x=0 to X=b x=b, as can be seen from Figure 1, the cylindrical shell sequence fills the entire object along the axis. So the total volume is DV DV volume element and-or integral
V=∫DV=∫2ΠXYDX=∫B02ΠXF (x) dx (2) \begin{equation} v=\int dv=\int 2\pi xydx=\int_0^b2\pi XF (x) dx\tag2 \end{equation}

Wherein Y=f (x) y=f (x), in principle, the volume V v can also be calculated by horizontal narrow band of the horizontal disk, but we will find it very difficult, because the given equation y=f (x) y=f (x) cannot use Y y to represent x x.


Figure 1

As with other integration applications, equation (1) (2) turns complex processes involving and limiting into concise expressions, and for the sake of clarity we ignore the details of the process.

As before, we recommend that you do not rote the formula (2). This formula is similar to the corresponding disk formula, and if it is only rote without thinking, it is easy to mix and type confidently. A better way is to draw, build (1) directly from the information visible in the diagram, and then integrate the form (2). In addition, this method has a greater advantage, we do not rely on any particular symbol, it is easy to apply the basic ideas to the various axes to rotate the object.

Example 1: In the previous article we calculated the volume of the sphere using the disk method. Now we are using the cylindrical shell method to solve this problem (Figure 2). The volume of the shell shown in the figure is