The curve integral of §10.1 to arc length

I. Introduction of the Concept

Assuming that there is a curve in the surface of the arc quality, at the beginning of the line density, and in the continuous, and the arc is the endpoint, now calculate the quality of the arc.

Insert a point arbitrarily on the

will be divided into a small arc segment. For the first small arc segment, because the line density function is on the continuous, when the length of the small arc segment is full hours, its mass is approximately equal to

Thus, the mass approximation of the entire curve arc is

With the largest of the length of this small arc, i.e.

In order to obtain the exact value of the mass, simply take the limit of the above and the type,

That is (1)

Putting aside the physical meaning of the above example, we introduce the concept of curve integral for arc length.

"Definition" is a smooth curved arc within the plane, where the function is bounded and the insertion point is arbitrarily

It divides into a small arc segment, set the length of the first small paragraph, for taking up a point, remember

Make and type

If the limit exists,

This limit is called a function in the curve arc on the arc length of the curve Integral, recorded.

i.e.

Where: called Integrand, called Integral arc segment.

Note:

1. The domain of the integrand is defined as all points on.

2, the above definition can be similarly extended to the situation of the space curve,

Set is a smooth curve of space, the function is bounded, then

3, if a closed curve, the general will be recorded as.

The properties of the curve integral of the arc length

Using the curve integral definition of arc length, we can prove the following properties

1,

2, if the constant,

3,

4, if the above, the

5, if, then

None of the above is proved and interested students can consult the books.

Three, the calculation method of the integral of arc length curve

Assumption curve by parametric equation

And the function is in the first order continuous derivative, the function is continuous on the top, and when the parameter is changed, the curve is traced according to the direction of point to point.

Take a series of points on the

Set them to correspond to a column of monotonically increasing parameter values

According to the definition

Here, and the point that corresponds to the parameter value

The

The calculation formula of arc length and the mean value theorem of definite integral are

Thus

(2)

Because the function is in the continuous, at the time, the length between the cells. Then on,

And

There is only one higher order infinitesimal, so we can replace the right end of the (2) type with

The limit of the right end and the type is the definite integral of the function on the interval. Because the function is continuous, so the definite integral exists, therefore, the upper-left curve integral also exists, and has

(3)

It is emphasized that the limit of the definite integral in the (3) type must be less than the upper limit, and the reason is

(2) The expression in the formula

Given, due to the length of the small arc segment, thus

So

Using the (3) formula, we can derive the following several curve integral formulas for arc length

1, the curve by the equation

When given,

2, the curve by the equation

When given,

3, the spatial curve by the parametric equation

When given,

"Example 1" calculation, where is the circumference

"Solution One" can be transformed into parametric equation

"Solution Two" curve about the axis symmetry, set is above the axis of a branch, then the equation should be

And the integrand is on the axis even symmetry, so

"Example 2" calculates the radius of the arc of the center angle for the rotational inertia of its symmetric axis (setting the line density).

Solution: Establish a coordinate system as shown in the figure

The

and

So

The curve integral of §10.2 to coordinate

I. Introduction of the Concept

A particle is moved in the plane from the point along a smooth curved arc, and in the moving process the particle is subjected to a variable force

The function of which functions, in the continuous, is calculated by the work of the variable force.

Insert a point arbitrarily on the

will be divided into small arc segments, and the coordinates of the points are.

Due to smooth and very short, there are directed line segments available

To approximately replace it, which, respectively, is the projection on the axis.

And because the function, in the continuous, can use the force at any point

To approximately replace the variable force on the Small Arc section.

When a particle moves along a small arc segment, the power of the variable force can be approximately taken as

Thus

For the exact value to be obtained, simply make, (is the largest of the length of this small arc segment), take the limit of the above and the type.

That is (1)

(1) The limit of the right end and the type is another new type limit, for which we introduce the curve integral concept of the coordinate.

"Definition" is defined as a smooth curved arc with a direction from point to point in the plane, a function, bounded on, and used on the dot

will be divided into a small arc segment, set

,

is the largest of the length of this little arc.

Any pick point

If the limit exists, then this limit is called a function in the curved arc of the curve integral to the coordinates, recorded.

Similarly, if the limit exists, then this limit is called a function in the curved arc of the curve integral to the coordinates, and recorded.

That

Where: called Integrand, called Integral arc segment.

Note:

1, to coordinate the curve integral is the direction of the Arc section on the axis of the projection, its value can be positive can also be negative. This is different from the constant positive value of the curve integral of the arc length.

2, the application often appears

This form, in the future, can be denoted into

Thus, the work of the variable force along the direction curve can be

3. The above definition can be generalized to the case that the integral curve arc is a spatially curved arc

And can be denoted into form

4. The existence theorem of curve integral to coordinate

If, in the direction of the smooth curve arc continuous, then

,

All exist.

This theorem can be similarly extended to the case of spatial curves.

The property of the integral of the coordinate curve

1. If it is determined by the direction in which the direction is divided and,

2, set is a curved arc, but the opposite direction of the curved arc, then

This property shows that the curve integral of the coordinate should pay special attention to the direction of Integral curve arc.

3, if, is a constant, then

The integral calculation method of coordinate curve

Theorem

is defined and continuous on a curved arc;

The parametric equation of the curve is

When the parameters are monotonically changed to, the point from the starting point along the movement to the endpoint;

function, which has a first-order continuous derivative at the interval of the endpoint, and

Then the curve integral exists, and

(4)

Proof: Insert a series of points arbitrarily (in the direction of adherence)

They correspond to the parameter values of the

The parameter values of this column are monotonically varying.

According to the definition of the curve integral of the coordinate

If the key is corresponding to the parameter value, then it should be between the and

and

Here, and in between with.

So

Because a function is contiguous on a closed interval (or), it can be replaced by the upper type, thereby

and equivalent to, so

The upper-right end and the type limit are definite integrals.

Because of the continuous, this definite integral exists, so the curve integral on the upper left side also exists, and has

Empathy can be verified