“Quadrature”（通常翻译成求积）并不完全等同于“Integral”这个术语在现代数学语言中的用法。以下是详细解释：

积分：这个概念指的是寻找曲线下的面积、表面下的体积，或者更广泛地，一个区间内数量的累积。Quadrature：这个术语与寻找一个正方形的面积历史上有关。“Quad"与“四”相关，平方某个数就是求其面积。在早期数学中，求解曲线下面积（积分的核心概念之一）涉及到用正方形或矩形来逼近，这个过程被称为求积。随着时间的推移，积分概念变得更加广泛，但“Quadrature”这个术语在某些情境下仍然保留下来：

数值求积（Numerical Quadrature）：这指的是使用数值方法近似计算积分的特定技术，比如将曲线下的面积划分成矩形并求和它们的面积。因此，“Quadrature”更像是一种执行积分的特定方式（通过方块逼近），而不是整个概念的完美同义词。

在现代数学课程中，你可能会比“Quadrature”更频繁地遇到“Integral”。

原文：

“Quadrature” isn’t quite a perfect synonym for “integral” in today’s mathematical language. Here’s the breakdown: Integral: This refers to the general concept of finding the area under a curve, the volume under a surface, or more generally, the accumulation of a quantity over an interval. Quadrature: This term has a historical connection to finding the area of a square. “Quad” relates to “four,” and squaring something is finding its area. In early mathematics, finding the area under a curve (a core concept of integration) involved approximating it with squares or rectangles. This process was called quadrature. Over time, the concept of integration became more general, but the term “quadrature” stuck around in some contexts:

Numerical Quadrature: This refers to specific techniques for approximating integrals using numerical methods like dividing the area under the curve into rectangles and summing their areas. So, “quadrature” is more like a specific way (approximation with squares) of performing integration, rather than a perfect synonym for the entire concept.

In modern mathematics courses, you’ll likely encounter “integral” more frequently than “quadrature.”