definite integral

简明释义

定积分

英英释义

例句

1.To analyze the total revenue, the accountant calculated the definite integral of the revenue function over the sales period.

为了分析总收入，会计师计算了销售期间收入函数的定积分。

2.In physics, the work done by a force can be determined using the definite integral of the force over a distance.

在物理学中，力所做的功可以通过力在一定距离上的定积分来确定。

3.The definite integral of a velocity function gives the total displacement over a given time interval.

速度函数的定积分给出了在给定时间间隔内的总位移。

4.The definite integral helps in calculating probabilities in statistics by finding the area under the probability density function.

在统计学中，定积分通过找到概率密度函数下的面积来帮助计算概率。

5.To find the area under the curve, we need to calculate the definite integral of the function from a to b.

要找到曲线下方的面积，我们需要计算从a到b的定积分。

作文

The concept of the definite integral is fundamental in calculus and plays a crucial role in various applications, including physics, engineering, and economics. In essence, a definite integral represents the accumulation of quantities, such as areas under curves, over a specified interval. Understanding this concept not only enhances our mathematical skills but also deepens our comprehension of the world around us.To begin with, the definite integral can be visualized geometrically. Imagine a continuous function graphed on a coordinate plane. The area between the curve of the function and the x-axis over a closed interval [a, b] can be calculated using the definite integral. This area is significant because it provides valuable information about the behavior of the function across that interval. For instance, if the function represents velocity, the definite integral of that function from time a to time b would yield the total distance traveled during that time period.Mathematically, the definite integral is denoted as follows: ∫[a, b] f(x) dx, where f(x) is the function being integrated, and a and b are the limits of integration. The process of calculating a definite integral involves finding the antiderivative of the function, which is then evaluated at the upper and lower limits. This procedure is often referred to as the Fundamental Theorem of Calculus, which connects differentiation and integration in a profound way.Moreover, the applications of the definite integral extend far beyond mere area calculations. In physics, for example, it is used to determine quantities like work done by a force or electric charge accumulated over time. In economics, the definite integral can help calculate consumer surplus or producer surplus, providing insights into market efficiency and welfare.One practical application of the definite integral can be observed in the analysis of motion. If we have a function that describes the acceleration of an object over time, we can use the definite integral to find the total change in velocity over a specific time interval. This is particularly useful in scenarios involving variable acceleration, where traditional methods of calculating distance might fall short.Additionally, the definite integral is instrumental in probability theory. The area under a probability density function (PDF) over a certain range gives the probability that a random variable falls within that range. This application highlights the importance of the definite integral in fields that rely heavily on statistical analysis and decision-making.In conclusion, the definite integral is not just a mathematical tool; it is a gateway to understanding complex concepts in various disciplines. Its ability to represent accumulation and provide insights into real-world phenomena makes it an invaluable part of both mathematics and applied sciences. By mastering the definite integral, one can unlock a deeper appreciation for the interconnectedness of different fields and the underlying principles that govern them. As we continue to explore and apply this concept, we enhance our capacity to analyze, interpret, and navigate the complexities of the world around us.

在微积分中，定积分的概念是基础，并在物理、工程和经济等多个应用领域中发挥着重要作用。简而言之，定积分表示在指定区间内数量的累积，例如曲线下方的面积。理解这一概念不仅提高了我们的数学技能，还加深了我们对周围世界的理解。首先，定积分可以通过几何图形进行可视化。想象一个连续函数在坐标平面上的图像。在闭区间[a, b]内，函数曲线与x轴之间的面积可以通过定积分计算得出。这个面积很重要，因为它提供了关于函数在该区间内行为的有价值信息。例如，如果函数代表速度，那么从时间a到时间b的定积分将得出在该时间段内行驶的总距离。在数学上，定积分表示为：∫[a, b] f(x) dx，其中f(x)是被积分的函数，a和b是积分的上下限。计算定积分的过程涉及找到函数的反导数，然后在上下限处进行求值。这个过程通常被称为微积分基本定理，它以深刻的方式将微分和积分联系在一起。此外，定积分的应用远不止于简单的面积计算。在物理学中，它用于确定力做功或电荷随时间累积等量。在经济学中，定积分可以帮助计算消费者剩余或生产者剩余，为市场效率和福利提供见解。定积分的一个实际应用可以在运动分析中观察到。如果我们有一个描述物体随时间加速度的函数，我们可以利用定积分来找出在特定时间区间内速度的总变化。这在涉及变加速的场景中特别有用，传统的计算距离的方法可能会失效。此外，定积分在概率论中也至关重要。某一概率密度函数（PDF）在某个范围内的面积给出了随机变量落在该范围内的概率。这一应用突显了定积分在依赖统计分析和决策制定的领域中的重要性。总之，定积分不仅仅是一个数学工具；它是理解各个学科复杂概念的入口。它能够表示累积并提供对现实世界现象的洞察，使其成为数学和应用科学中不可或缺的一部分。通过掌握定积分，我们可以解锁对不同领域相互关联性及其背后原则的更深层次的欣赏。随着我们继续探索和应用这一概念，我们增强了分析、解释和驾驭周围世界复杂性的能力。