Name: definite integrals: a step-by-step strategy
Uploaded: 2026-01-12T12:24:28.000Z
Duration: 1 min 23 s
Channel: JoVE

考虑一个任意函数的定积分，该积分表示曲线下的面积。如果下限固定，而上限被赋予一个变量，则所得的定积分将表现为关于变量x的函数g(x)，该函数称为累积函数。当x变化时，曲线下方的面积也随之变化。g(x) 的导数展示了面积如何变化，这可以通过回到极限定义来求得。在此，g(x + h) − g(x) 表示曲线在 x 与 x + h 之间下方一个细垂直条带的面积。当该条带无限变窄时，其面积可近似为一个宽度为 h、高度为 f(x) 的矩形。那么其导数就是被积函数 f(x)。这被称为微积分基本定理的第一种形式。例如，在一个串联 RC 电路中，电容器在任意时刻 t 所带的电荷量由该积分表示。电容器中的电流，即其电荷的瞬时变化率，根据微积分基本定理，等于被积函数中的表达式。

定积分的性质：问题求解

Fundamental Theorem of Calculus I

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Integrals

积分

KEY TERMS AND CONCEPTS

关键术语与概念

definite integrals: a step-by-step strategy

Definite integrals are solved best with a clear, step-by-step strategy. The first step is to identify what the problem is asking for. The quantity may be accumulation, area, force, or probability. It is also important to note the integration limits and any constraints that affect the setup.

Before evaluating the integral, look for properties that can simplify the work. If the expression has multiple parts, break it into smaller integrals. Known values from earlier work may help too. Symmetry can also reduce the amount of calculation needed.

Comparison ideas can help when an integral is related to another function that is always larger or smaller. In that case, you can use bounds to estimate the result. Once the integral is ready, choose the most suitable method for the problem. If an exact answer is not possible, approximation may be the better choice.

After solving, interpret the result in context. A value for area should make sense as non-negative, and a total accumulation should be reasonable for the situation. The final step is to check the work. Review the solution for errors, try another method if possible, and confirm the answer when you can.

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key terms and concepts

Learn a clear method for solving definite integrals by identifying the quantity, using symmetry or bounds, choosing a method, and checking the result.

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calculus

Definite Integrals: A Step-by-Step Strategy

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