∭ 三重积分计算器（数值）Triple Integral Calculator (Numerical)
数值计算三重积分 ∭ f(x, y, z) dz dy dx：x 的上下限为常数，y 的上下限可以是 x 的函数，z 的上下限可以是 x、y 的函数，使用高斯-勒让德求积，适合计算体积、质量和重心。Evaluate a triple integral ∭ f(x, y, z) dz dy dx numerically: constant x limits, y limits that may depend on x, and z limits that may depend on x and y, using Gauss–Legendre quadrature — useful for volumes, masses and centroids.
使用说明与常见问题How-to & FAQ
「三重积分计算器（数值）」是什么？What is the Triple Integral Calculator (Numerical) tool?
数值计算三重积分 ∭ f(x, y, z) dz dy dx：x 的上下限为常数，y 的上下限可以是 x 的函数，z 的上下限可以是 x、y 的函数，使用高斯-勒让德求积，适合计算体积、质量和重心。Evaluate a triple integral ∭ f(x, y, z) dz dy dx numerically: constant x limits, y limits that may depend on x, and z limits that may depend on x and y, using Gauss–Legendre quadrature — useful for volumes, masses and centroids.
如何使用「三重积分计算器（数值）」？How do I use the Triple Integral Calculator (Numerical) tool?
在「三重积分计算器（数值）」中，填写「被积函数 f(x, y, z)」、「x 下限」、「x 上限」、「y 下限（可含 x）」、「y 上限（可含 x）」等参数，结果会随输入实时计算并显示在下方。In the Triple Integral Calculator (Numerical): Fill in "Integrand f(x, y, z)", "x lower", "x upper", "y lower (may use x)", "y upper (may use x)"; results are recalculated instantly as you type.
积分顺序是什么？What is the order of integration?
按 ∫ₓ ∫ᵧ ∫_z f dz dy dx 计算：最内层对 z 积分（上下限可含 x、y），中间对 y（上下限可含 x），最外层对 x（上下限为常数）。∫ₓ ∫ᵧ ∫_z f dz dy dx: innermost over z (limits may use x and y), then y (limits may use x), then x (constant limits).
精度如何？How accurate is it?
每个方向使用 12、24 或 48 点高斯-勒让德求积，结果按所选点数计算，并显示与一半点数结果的差，作为粗略的收敛判断（不是真实误差）。被积函数和积分限都光滑时，通常可以精确到 10 位以上；积分限含根号（如球体，边界处导数无穷大）时收敛很慢，“很高”精度下单位球体积为 4.18881，与精确值 4π/3 ≈ 4.18879 相比只有约 5 位有效数字。可以省略乘号，如 2xyz 或 x(y+z)。Each direction uses 12-, 24- or 48-point Gauss–Legendre quadrature. The value is computed at the chosen count and shown with its difference from the half-count rule as a rough convergence check, not the true error. With smooth integrands and limits it is usually good to 10+ digits; limits with square roots (like a sphere, whose boundary has infinite slope) converge slowly, and at Very high the unit ball gives 4.18881 against the exact 4π/3 ≈ 4.18879, only about 5 significant digits. The multiplication sign can be left out, as in 2xyz or x(y+z).
可以算体积吗？Can it compute a volume?
可以：令被积函数为 1，积分值就是区域体积。例如单位球的体积为 4π/3 ≈ 4.18879。Yes — set the integrand to 1 and the integral is the region's volume; the unit ball gives 4π/3 ≈ 4.18879.
「三重积分计算器（数值）」免费吗？需要注册吗？Is the Triple Integral Calculator (Numerical) free? Do I need an account?
「三重积分计算器（数值）」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 Triple Integral Calculator (Numerical) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.