I. Introduction. A periodic function is a class of functions with special properties, and its value changes repeatedly over a certain period. The definite integral is the value of the function after integration in a certain interval, and is often used to solve problems such as the area, volume, and average value of the function. For periodic functions, their definite integrals also have special properties. In this article, we will introduce the calculation methods and applications of definite integrals of periodic functions in detail.
2. Definition and properties of periodic functions.
A periodic function is one in which a positive number t exists for any real number t in its defined domain, such that for all x's there is f(x+t)=f(x). Common periodic functions are sine function, cosine function, tangent function, etc. Periodic functions have some special properties, such as:
1.The image of the periodic function is closed, i.e., in a period, the value of the function goes from the maximum to the minimum to the maximum, forming a closed loop.
2.The sum and difference of the periodic function are also periodic functions.
3.The derivatives and integrals of the periodic function are also periodic functions.
3. The method of calculating the definite integral of the periodic function.
For the periodic function f(x), the definite integral over the interval [a,b] can be expressed as (f(x))dx. Since f(x) is a periodic function, the image of f(x) is a closed curve in one period. Thus, on the interval [a,b], the image of f(x) is also a closed curve, which can be divided into several small closed curve segments, and the value of f(x) on each segment is constant.
Therefore, when calculating the definite integral of a periodic function, the interval [a,b] can be divided into several intervals, and the integral on each intervals is a constant multiplied by the length of the interval. Finally, the integrals between all cells are added together to obtain the value of the definite integral.
For example, for the sinusoidal function sin(x), the definite integral over the interval [0,2 ] can be expressed as (sin(x))dx. The interval can be divided into several intervals, and the integrals on each intervals are -cos(x)+c (c is a constant), and finally the integrals between all cells can be added to obtain the value of the definite integral.
4. The properties and applications of definite integrals of periodic functions.
The definite integral of a periodic function has some special properties, such as:
1.For any real number t, (f(x+t))dx= (f(x))dx.
2.If f(x) is an even function, then (-f(x))dx=0;If f(x) is an odd function, then (-f(x))dx=2 (f(x))dx.
3.If f(x) is a periodic function and the period is t, then for any real number t, there is ((f(x+t))dx= (f(x)))dx.
The definite integral of the periodic function has a wide range of applications, such as in electronic engineering, mechanical engineering and other fields. For example, in electronic engineering, definite integrals can be used to solve for currents and voltages for alternating currentsIn mechanical engineering, definite integrals can be used to solve for the length and area of curves, etc.
V. Conclusions. This paper introduces the definition and properties of periodic functions, as well as the calculation methods and applications of definite integrals of periodic functions. Through the introduction of this paper, it can be found that the definite integral of periodic function is a class of functions with special properties, and its calculation methods and applications have certain difficulty and complexity. Therefore, in practical application, it is necessary to choose the appropriate calculation method and formula to calculate the value of the definite integral according to the specific situation.