double integral

简明释义

重积分

英英释义

例句

1.When dealing with functions of two variables, a double integral (双重积分) is essential for integration.

在处理两个变量的函数时，double integral（双重积分）对于积分是必不可少的。

2.In physics, the double integral (双重积分) can be used to find the mass of a two-dimensional object.

在物理学中，double integral（双重积分）可以用来找到二维物体的质量。

3.We applied a double integral (双重积分) to compute the total charge in a given area.

我们应用了一个double integral（双重积分）来计算给定区域内的总电荷。

4.The area of a region can be determined by evaluating a double integral (双重积分) over that region.

一个区域的面积可以通过对该区域进行计算的double integral（双重积分）来确定。

5.To calculate the volume under a surface, we often use a double integral (双重积分).

为了计算曲面下的体积，我们通常使用一个double integral（双重积分）。

作文

In the realm of calculus, the concept of a double integral is essential for understanding how to compute volumes and areas in higher dimensions. A double integral, as the name suggests, involves integrating a function of two variables over a specific region in the xy-plane. This process allows us to calculate quantities that are not easily determined by single-variable integration alone. For instance, if we want to find the volume under a surface defined by a function f(x, y) over a rectangular region in the xy-plane, we would use a double integral. To perform a double integral, one must first define the limits of integration, which correspond to the boundaries of the region over which we are integrating. These limits can be constants or functions themselves, depending on the shape of the region. The double integral is typically expressed as:∬_R f(x, y) dA,where R represents the region of integration, and dA is the differential area element, often expressed as dx dy or dy dx. The order of integration can be switched based on the problem's requirements, but it is crucial to ensure that the limits are adjusted accordingly.Understanding the geometric interpretation of a double integral can significantly enhance our comprehension of its applications. When visualizing the double integral, think of it as summing up infinitely small pieces of volume above the region R. Each infinitesimal area element contributes to the total volume, and by integrating over the entire region, we obtain the desired result.One common application of double integrals is in calculating the area of irregular shapes. By defining the function f(x, y) as 1, the double integral simplifies to finding the area of the region R. This technique is particularly useful in fields like physics and engineering, where understanding the properties of complex shapes is essential.Moreover, double integrals can also be employed in probability theory. When dealing with continuous random variables, the joint probability density function can be integrated over a specified region to determine the probability of the variables falling within that area. This application highlights the versatility of double integrals beyond mere geometry.Despite their usefulness, double integrals can sometimes pose challenges, especially when the region of integration is complicated. In such cases, changing to polar coordinates can simplify the process. The transformation involves expressing x and y in terms of r and θ, leading to the following form:∬_R f(r cos(θ), r sin(θ)) r dr dθ,where r is the distance from the origin and θ is the angle. This substitution can make evaluating the double integral more manageable, particularly for circular or annular regions.In conclusion, mastering the concept of a double integral is vital for anyone studying calculus or related fields. Its applications in computing volumes, areas, and probabilities demonstrate its significance in both theoretical and practical contexts. As we continue to explore the intricacies of mathematics, the double integral remains a powerful tool that helps us understand and quantify the world around us.

在微积分的领域中，双重积分的概念对于理解如何计算高维空间中的体积和面积至关重要。正如其名称所暗示的，双重积分涉及对一个二维变量的函数在xy平面上的特定区域进行积分。这个过程使我们能够计算那些无法仅通过单变量积分轻易确定的量。例如，如果我们想要找出在由函数f(x, y)定义的曲面下方的体积，我们将使用双重积分。进行双重积分时，首先必须定义积分的限制，这些限制对应于我们正在积分的区域的边界。这些限制可以是常数，也可以是函数，具体取决于区域的形状。双重积分通常表示为：∬_R f(x, y) dA，其中R表示积分区域，dA是微分面积元素，通常表示为dx dy或dy dx。根据问题的要求，可以改变积分的顺序，但确保相应地调整限制是至关重要的。理解双重积分的几何解释可以显著增强我们对其应用的理解。在想象双重积分时，可以将其视为对区域R上方的无限小体积块进行求和。每个微小的面积元素都对总体积有所贡献，通过对整个区域进行积分，我们获得所需的结果。双重积分的一个常见应用是在计算不规则形状的面积时。通过将函数f(x, y)定义为1，双重积分简化为寻找区域R的面积。这种技术在物理和工程等领域特别有用，因为理解复杂形状的性质至关重要。此外，双重积分也可以用于概率理论。当处理连续随机变量时，可以在指定区域内对联合概率密度函数进行积分，以确定变量落在该区域内的概率。这一应用突显了双重积分在几何学之外的多样性。尽管双重积分非常有用，但有时也会带来挑战，尤其是当积分区域复杂时。在这种情况下，转换为极坐标可能会简化过程。这个转换涉及将x和y表示为r和θ，从而得到以下形式：∬_R f(r cos(θ), r sin(θ)) r dr dθ，其中r是距离原点的距离，θ是角度。这个替换可以使评估双重积分变得更加可管理，特别是对于圆形或环形区域。总之，掌握双重积分的概念对于任何学习微积分或相关领域的人来说都是至关重要的。它在计算体积、面积和概率方面的应用展示了它在理论和实践背景下的重要性。在我们继续探索数学的复杂性时，双重积分仍然是一种强有力的工具，帮助我们理解和量化周围的世界。