The application of-chape6-definite integral in "University calculus"--seeking volume

Definite integral a broad application is in the solution of some "seemingly irregular" geometry of the volume, the reason seems to be irregular, because there is a certain "rule" under the rule, we just need to grasp these clues to achieve the integral operation of the volume.

Method 1: Slicing method.

Here because the method of processing thought and the typical discrete Riemann and to the process of continuous integration is similar, so here is no longer repeated derivation, directly gives how to apply and examples.

Based on this theorem, we can introduce the Cavalieri principle directly. The Cavalieri principle shows that the same volume of geometry with the same cross-sectional area at each height is the same, and intuitively understood, like these two stacks of "stacked coin" graphs.

Let's look at some examples below.

The computational volume of this method has a more stringent limit, the most important of which is that the cross section A (x) of this irregular geometry is a typical geometry, and this method is used to calculate the volume that can be summed up as the following algorithm flow:

1. Draw three-dimensional and typical cross-sectional sketches
2. Typical cross sectional area a (x)
3. To find an integral limit
4. Using the basic theorem of calculus to find the integral of a (x)

So for this example, based on the sketch, we can list the following definite integral formula:

Take a look at a slightly more challenging diagram.

The choice of cross-section is tricky, and a key principle is that the cross section needs to remain a typical planar shape when the cross section of the selection is moved along the axis of the integral variable. The x is still used as the integral variable, then the cross section A (x) is a rectangle, so we can list the following formulas:

Method 2:

A disc method for handling a rotating body.

In fact, this method is essentially a slicing method to deal with rotating body. Before introducing this method, it is necessary to introduce what is a rotating body: the space geometry that rotates a planar shape around the axis for one week is what we call a rotating body.

The rotating body is very regular to follow, assuming that we rotate the curve f (x) and the x-axis of the curved edge ladder, we can find that the x as an integral variable, cross section is a typical shape-circle, and the radius can be given by the curve f (x).

Let's look at a few examples.

It is easy to list the following definite integral formula.

The application of-chape6-definite integral in "University calculus"--seeking volume