∬ 二重积分计算器（矩形或 y 介于两曲线之间的区域）Double Integral Calculator (Rectangle or Region Between Two Curves)
输入被积函数 f(x, y)（如 x*y、x^2 + y^2、2），在矩形区域 [x₀, x₁] × [y₀, y₁] 上，或在 x 从 a 到 b、y 从 g₁(x) 到 g₂(x) 的区域上，用二维辛普森法数值计算二重积分，同时给出区域面积和平均值。本地处理。Enter an integrand f(x, y) (such as x*y, x^2 + y^2 or 2) and integrate it numerically with 2D Simpson rule over a rectangle [x₀, x₁] × [y₀, y₁] or over the region where x runs from a to b and y from g₁(x) to g₂(x), with the region area and the average value. Processed locally.
使用说明与常见问题How-to & FAQ
「二重积分计算器（矩形或 y 介于两曲线之间的区域）」是什么？What is the Double Integral Calculator (Rectangle or Region Between Two Curves) tool?
输入被积函数 f(x, y)（如 x*y、x^2 + y^2、2），在矩形区域 [x₀, x₁] × [y₀, y₁] 上，或在 x 从 a 到 b、y 从 g₁(x) 到 g₂(x) 的区域上，用二维辛普森法数值计算二重积分，同时给出区域面积和平均值。本地处理。Enter an integrand f(x, y) (such as x*y, x^2 + y^2 or 2) and integrate it numerically with 2D Simpson rule over a rectangle [x₀, x₁] × [y₀, y₁] or over the region where x runs from a to b and y from g₁(x) to g₂(x), with the region area and the average value. Processed locally.
如何使用「二重积分计算器（矩形或 y 介于两曲线之间的区域）」？How do I use the Double Integral Calculator (Rectangle or Region Between Two Curves) tool?
在「二重积分计算器（矩形或 y 介于两曲线之间的区域）」中，填写「区域」、「被积函数 f(x, y)」、「x 下限」、「x 上限」、「y 下限」等参数，结果会随输入实时计算并显示在下方。In the Double Integral Calculator (Rectangle or Region Between Two Curves): Fill in "Region", "Integrand f(x, y)", "x lower limit", "x upper limit", "y lower limit"; results are recalculated instantly as you type.
二重积分表示什么？What does a double integral represent?
∬ f(x, y) dA 是函数在平面区域上的累积量。f = 1 时等于区域面积；f ≥ 0 时可以理解为曲面 z = f(x, y) 与区域之间的体积。∬ f(x, y) dA accumulates f over a region of the plane. With f = 1 it is the area; with f ≥ 0 it is the volume between the surface z = f(x, y) and the region.
曲线边界怎么填？How do I enter curved boundaries?
选择“y 介于 g₁(x) 与 g₂(x) 之间”，x 从 a 到 b，y 的下界和上界写成 x 的表达式，例如抛物线 x^2 与直线 x 围成的区域：x 从 0 到 1，g₁ = x^2，g₂ = x。Choose y between g₁(x) and g₂(x), give x from a to b, and write the lower and upper y bounds as expressions in x. For the region between x^2 and x: x from 0 to 1, g₁ = x^2, g₂ = x.
是精确值吗？Is the result exact?
不是符号积分，而是数值积分。多项式等光滑函数结果非常接近精确值；在区域内不连续或趋于无穷的函数可能不准确。It is a numerical, not symbolic, integral. Smooth functions such as polynomials come out very close to exact; functions that jump or blow up inside the region may be inaccurate.
能举个计算实例吗？Can you show a worked example?
例如在「二重积分计算器（矩形或 y 介于两曲线之间的区域）」中填 区域 = 矩形，被积函数 f(x, y) = 2，x 下限 = 0，x 上限 = 3，y 下限 = 0，y 上限 = 4，结果区显示：∬ f dA：24；区域面积（有向）：12；平均值 f̄：2；ℹ 二维辛普森法数值结果，光滑函数通常精确到 8 位左右。上限小于下限时结果带负号。For example, entering Region = Rectangle, Integrand f(x, y) = 2, x lower limit = 0, x upper limit = 3, y lower limit = 0, y upper limit = 4 in the Double Integral Calculator (Rectangle or Region Between Two Curves) gives: ∬ f dA: 24; Region area (signed): 12; Average value f̄: 2; ℹ 2D Simpson rule; usually about 8 digits for smooth functions. Limits given in reverse order flip the sign.
哪些输入会被检查？Which inputs are validated?
「二重积分计算器（矩形或 y 介于两曲线之间的区域）」会检查这些输入——「x 下限」、「y 下限」：所选方式用到时必须填写；「x 上限」、「y 上限」：可留空；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Double Integral Calculator (Rectangle or Region Between Two Curves) checks its inputs — "x lower limit", "y lower limit": required when the chosen option uses it; "x upper limit", "y upper limit": optional; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.
有哪些选项可以调整？What options can I adjust?
「二重积分计算器（矩形或 y 介于两曲线之间的区域）」的设置：「区域」可选 矩形、y 介于 g₁(x) 与 g₂(x) 之间（默认 矩形）；「被积函数 f(x, y)」默认 2；「x 下限」默认 0；「x 上限」默认 3；「y 下限」默认 0；「y 上限」默认 4；「g₁(x)（下边界）」默认 0；「g₂(x)（上边界）」默认 x。每次生成结果都按当前设置处理。Settings in the Double Integral Calculator (Rectangle or Region Between Two Curves) — "Region": Rectangle, y between g₁(x) and g₂(x) (default Rectangle); "Integrand f(x, y)" default 2; "x lower limit" default 0; "x upper limit" default 3; "y lower limit" default 0; "y upper limit" default 4; "g₁(x) (lower)" default 0; "g₂(x) (upper)" default x. The result always uses the current settings.
计算精度如何？How precise are the results?
「二重积分计算器（矩形或 y 介于两曲线之间的区域）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Double Integral Calculator (Rectangle or Region Between Two Curves) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「二重积分计算器（矩形或 y 介于两曲线之间的区域）」免费吗？需要注册吗？Is the Double Integral Calculator (Rectangle or Region Between Two Curves) free? Do I need an account?
「二重积分计算器（矩形或 y 介于两曲线之间的区域）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Double Integral Calculator (Rectangle or Region Between Two Curves) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.