Applications involving the integral as a sum | 积分作为求和的应用

📚 Applications involving the integral as a sum | 积分作为求和的应用

The integral was originally developed as a way of summing infinitely many infinitesimally small quantities. In AS Mathematics, this idea is central to applications such as finding areas, volumes, displacement, and total accumulated change.

积分最初是为了累加无穷多个无穷小量而发展起来的。在AS数学中，这一思想是求面积、体积、位移和累积变化等应用的核心。

1. The Definite Integral as a Limit of Sums | 定积分作为求和的极限

The definite integral ∫_a^b f(x) dx is defined as the limit of a Riemann sum: lim_n→∞ Σ_i=1ⁿ f(x_i*) Δx, where Δx = (b − a)/n. This represents the net accumulation of f(x) over [a, b].

定积分 ∫_a^b f(x) dx 被定义为黎曼和的极限：lim_n→∞ Σ_i=1ⁿ f(x_i*) Δx，其中 Δx = (b − a)/n。它表示 f(x) 在区间 [a, b] 上的净累积量。

sum: area when f(x) ≥ 0, signed area when f(x) can be negative, volume when summing cross‑sectional disks, or total distance when summing speed over time.

积分的每一个应用都可以解释为这种求和：当 f(x) ≥ 0 时为面积，当 f(x) 可正可负时为带符号面积，累加截面圆盘时得到体积，或累加速度随时间变化时得到总路程。

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2. Area Under a Curve (x‑axis) | 曲线下的面积（x轴）

y = f(x) is continuous and f(x) ≥ 0 for a ≤ x ≤ b, the area A between the curve and the x‑axis is given by A = ∫_a^b f(x) dx.

若 y = f(x) 在 a ≤ x ≤ b 上连续且 f(x) ≥ 0，则曲线与 x 轴之间的面积 A = ∫_a^b f(x) dx。

s of height f(x) and width dx. For example, the area under y = x² from x = 0 to x = 3 is ∫₀³ x² dx = [x³/3]₀³ = 9.

积分累加了高度为 f(x)、宽度为 dx 的无穷多个竖条的面积。例如，y = x² 在 x = 0 到 x = 3 下的面积为 ∫₀³ x² dx = [x³/3]₀³ = 9。

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3. Area Between a Curve and the y‑axis | 曲线与y轴之间的面积

When a curve is expressed as x = g(y), the area between the curve and the y‑axis from y = c to y = d is ∫_c^d g(y) dy. This uses horizontal strips of length g(y) and thickness dy.

当曲线表示为 x = g(y) 时，从 y = c 到 y = d 曲线与 y 轴之间的面积为 ∫_c^d g(y) dy。这里使用了长度为 g(y)、厚度为 dy 的水平条带。

For instance, the area bounded by y = √x, the y‑axis, and y = 2 is found by rewriting as x = y² and integrating from y = 0 to 2: ∫₀² y² dy = 8/3.

例如，由 y = √x、y 轴和 y = 2 围成的面积可以通过改写为 x = y² 并对 y 从 0 到 2 积分得到：∫₀² y² dy = 8/3。

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4. Area Between Two Curves | 两曲线之间的面积

If f(x) ≥ g(x) on [a, b], the area between the curves is ∫_a^b [f(x) − g(x)] dx. The integral sums the differences of the two functions at each x, giving the vertical gap.

如果在 [a, b] 上 f(x) ≥ g(x)，则两曲线之间的面积为 ∫_a^b [f(x) − g(x)] dx。积分累加了每个 x 处两个函数的差，从而得到垂直间隙的面积。

split the interval where one is above the other and compute the absolute difference: A = ∫_a^b |f(x) − g(x)| dx.

当曲线相交多次时，需将区间按上下关系分割并计算绝对差：A = ∫_a^b |f(x) − g(x)| dx。

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5. Volume of Revolution About the x‑axis | 绕x轴的旋转体体积

When a region under y = f(x) from x = a to x = b is revolved about the x‑axis, the solid's volume is V = π ∫_a^b [f(x)]² dx. Each infinitesimal disk has radius f(x) and thickness dx, so its volume is π [f(x)]² dx.

当 y = f(x) 下方从 x = a 到 x = b 的区域绕 x 轴旋转时，所得立体的体积为 V = π ∫_a^b [f(x)]² dx。每个无穷小圆盘的半径为 f(x)，厚度为 dx，因此其体积为 π [f(x)]² dx。

Example: rotating y = 2x from x = 0 to x = 1 gives V = π ∫₀¹ (2x)² dx = π ∫₀¹ 4x² dx = (4π/3).

例如：将 y = 2x 在 x = 0 到 1 之间旋转，得到 V = π ∫₀¹ (2x)² dx = π ∫₀¹ 4x² dx = (4π/3)。

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6. Volume of Revolution About the y‑axis | 绕y轴的旋转体体积

To revolve a region about the y‑axis, express x in terms of y: if x = g(y) from y = c to y = d, the volume is V = π ∫_c^d [g(y)]² dy. This sums disks perpendicular to the y‑axis.

要绕 y 轴旋转区域，需将 x 表示为 y 的函数：若 x = g(y) 从 y = c 到 y = d，则体积为 V = π ∫_c^d [g(y)]² dy。这是累加垂直于 y 轴的圆盘。

Care must be taken when the region is not bounded solely by the axes; subtract volumes of inner cylinders if a hollow region is formed.

当区域并非仅由坐标轴围成时，要注意减去内圆柱的体积（如果形成了中空部分）。

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7. Kinematics: Displacement and Total Distance | 运动学：位移与路程

velocity of a particle, the displacement over [t₁, t₂] is the integral of velocity: Δs = ∫_t₁^t₂ v(t) dt. This is the net change in position.

如果 v(t) 是粒子的速度，则在 [t₁, t₂] 内的位移是速度的积分：Δs = ∫_t₁^t₂ v(t) dt。这是位置的净变化。

The total distance travelled is the integral of the speed |v(t)|: D = ∫_t₁^t₂ |v(t)| dt. This sums all motion regardless of direction, eliminating negative contributions from backward movement.

总路程是速率 |v(t)| 的积分：D = ∫_t₁^t₂ |v(t)| dt。它累加了所有的运动，不论方向，消除了向后运动带来的负贡献。

Acceleration a(t) can also be integrated to give velocity: v(t) = ∫ a(t) dt + v₀.

加速度 a(t) 也可通过积分得到速度：v(t) = ∫ a(t) dt + v₀。

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8. Average Value of a Function | 函数的平均值

The average value of f(x) on [a, b] is f_avg = (1/(b−a)) ∫_a^b f(x) dx. This is the constant height that gives the same area as the function over the interval.

f(x) 在 [a, b] 上的平均值为 f_avg = (1/(b−a)) ∫_a^b f(x) dx。这是一个常数高度，它与函数在该区间上形成的面积相等。

ding by the length yields the mean. For example, the average of sin x on [0, π] is (1/π) ∫₀^π sin x dx = 2/π.

积分聚集了所有的函数值，除以区间长度即得到均值。例如，sin x 在 [0, π] 上的平均值为 (1/π) ∫₀

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