1. Calculating area and volume: Integration was originally introduced to solve practical problems, and one of the most important applications is to calculate the area and volume of plane and three-dimensional graphics. For example, the area of a circle and the volume of a cylinder can be calculated by definite integral.
2. Solve physical and engineering problems: In physics and engineering, many problems can be solved by integration. For example, solving the trajectory of an object under the action of gravity, the charge distribution in the circuit and so on.
3. Describe the properties of the function: Integral can be used to describe some important properties of the function, such as continuity and differentiability. For example, if a function is continuous in an interval, then its definite integral in the interval must exist; If a function is differentiable in an interval, its indefinite integral must exist in this interval.
4. Establish differential equation: differential equation is an important tool to describe natural phenomena, and integration is the basic method to establish differential equation. By integrating both sides of the differential equation at the same time, a new equation can be obtained, which is usually easier to solve.
5. Studying infinite series: Infinite series is an important concept in mathematical analysis, and integral can be used to study the properties of infinite series. For example, by comparing the integrals of two infinite series, we can judge whether they converge.
6. Develop probability theory and statistics: In probability theory and statistics, integrals are used to calculate probabilities and expectations. For example, by calculating the definite integral of a random variable, its probability density function can be obtained; By calculating the expected value of a random variable, its average value can be obtained.