Riemann Sums, Summation Notation, and Definite Integral Notation · 黎曼和、求和记号与定积分记号
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| summation notation/sʌˈmeɪʃn nəʊˈteɪʃn/ | 求和记号 | qiú hé jì hào |
| definite integral/ˈdefɪnət ˈɪntɪɡrəl/ | 定积分 | dìng jī fēn |
| limits of integration/ˈlɪmɪts ɒv ˌɪntɪˈɡreɪʃn/ | 积分限 | jī fēn xiàn |
Writing "add up all the rectangles" compactly
- A Riemann sum with many rectangles needs tidy notation — enter the summation notation 求和记号.
- $\displaystyle\sum_{i=1}^{n} f(x_i)\,\Delta x$ means "add $f(x_i)\,\Delta x$ for $i=1$ to $n$."
- Each strip has width $\Delta x=\dfrac{b-a}{n}$ and height $f(x_i)$ at a sample point.
- The $\Sigma$ just says "sum these rectangle areas."
用紧凑记号写"把所有矩形加起来"
- 有许多矩形的黎曼和需要整洁的记号——求和记号登场。
- $\displaystyle\sum_{i=1}^{n} f(x_i)\,\Delta x$ 表示"对 $i=1$ 到 $n$ 把 $f(x_i)\,\Delta x$ 加起来"。
- 每条宽 $\Delta x=\dfrac{b-a}{n}$,高为样本点处的 $f(x_i)$。
- $\Sigma$ 只是说"把这些矩形面积求和"。
For · 支持 $n$ equal strips on $[a,b]$, the width $\Delta x$ is... · 对于 $n$ 等分的 $[a,b]$，宽度 $\Delta x$ 是...
The interval width divided by the number of strips. · 区间宽度除以条带数量。
The symbol $\sum_{i=1}^{n}$ tells you to... · 符号 $\sum_{i=1}^{n}$ 告诉你去...
Sigma means sum over the index range. · Sigma 表示按索引范围求和。
Let the rectangles get infinitely thin
- The estimate improves as $n$ grows and $\Delta x$ shrinks. Take the limit $n\to\infty$.
- In that limit the rectangle sum becomes the exact area — no approximation left.
- That limiting value is the definite integral 定积分.
-
$$\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x$$
让矩形变得无限薄
- 当 $n$ 增大、$\Delta x$ 缩小,估计变好。取极限 $n\to\infty$。
- 在那个极限里,矩形和变成精确面积——不再有近似。
- 那个极限值就是定积分。
-
$$\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x$$
The integral is the exact area · 积分即为精确面积
y = x²
As the rectangles thin to nothing, their sum becomes the exact shaded area — the definite integral. · 当矩形变得无限薄时，它们的总和变为精确的阴影面积——即定积分。
The definite integral $\int_a^b f\,dx$ is the limit of a Riemann sum as... · 定积分 $\int_a^b f\,dx$ 是黎曼和在...时的极限
Infinitely many infinitely-thin rectangles → exact area. · 无穷多个无限薄的矩形 → 精确面积。
Reading the integral notation
- $\displaystyle\int_a^b f(x)\,dx$: the $\int$ is a stretched "S" for sum; $a$ and $b$ are the limits of integration 积分限.
- $f(x)$ is the integrand (what you're accumulating); $dx$ is the infinitely thin width.
- Read it "the integral of $f$ of $x$, from $a$ to $b$."
- It denotes the exact signed area between the curve and the $x$-axis over $[a,b]$.
读积分记号
- $\displaystyle\int_a^b f(x)\,dx$:$\int$ 是拉长的"S",代表求和;$a$ 和 $b$ 是积分限。
- $f(x)$ 是被积函数(你在累积的东西);$dx$ 是无限薄的宽度。
- 读作"$f$ 关于 $x$ 从 $a$ 到 $b$ 的积分"。
- 它表示曲线与 $x$ 轴之间在 $[a,b]$ 上的精确带符号面积。
In $\int_a^b f(x)\,dx$, the function $f(x)$ being accumulated is called the ____. · 在 $\int_a^b f(x)\,dx$ 中，被累积的函数 $f(x)$ 被称为 ____。
$a,b$ are the limits; $f(x)$ is the integrand. · $a,b$ 是上下限；$f(x)$ 是被积函数。
It's still signed area
- Just like the Riemann sum, the definite integral is signed.
- Where $f>0$ the integral adds area; where $f<0$ it subtracts.
- So $\int_a^b f\,dx$ can be positive, negative, or zero.
- $\Delta x$ carries the direction: going from $a$ to $b$ with $a
它仍是带符号面积
- 和黎曼和一样,定积分是带符号的。
- $f>0$ 之处积分加面积;$f<0$ 之处减面积。
- 所以 $\int_a^b f\,dx$ 可以为正、为负或为零。
- $\Delta x$ 带着方向:从 $a$ 到 $b$ 且 $a
A definite integral $\int_a^b f\,dx$ evaluates to a single number. · 定积分 $\int_a^b f\,dx$ 的结果是一个单一数值。
It is a signed-area number, not a function. · 它是一个有向面积的数值，而非函数。
If $f(x)<0$ over all of $[a,b]$, then $\int_a^b f\,dx$ is... · 如果 $f(x)<0$ 覆盖整个 $[a,b]$，则 $\int_a^b f\,dx$ 是...
Below-axis area is negative signed area. · x轴下方的面积为负有向面积。
The definite integral is a number (a signed area), not a function — don't confuse it with the indefinite integral (a family of antiderivatives, lesson 6.8). And $\int_a^b f\,dx$ is signed area: a curve dipping below the axis makes a negative contribution, so the integral is not the same as "total geometric area."
定积分是一个数(带符号面积),不是函数——别把它与不定积分(一族原函数,6.8 课)搞混。而且 $\int_a^b f\,dx$ 是带符号面积:曲线跌到轴下方会做出负贡献,所以积分不等同于"几何总面积"。
Express the area under $f(x)=x^2$ on $[0,2]$ as a definite integral.
- Width of each of $n$ strips: $\Delta x=\dfrac{2-0}{n}=\dfrac2n$.
- Riemann sum: $\displaystyle\sum_{i=1}^{n} f(x_i)\,\Delta x$.
- Exact area: $\displaystyle\int_0^2 x^2\,dx=\lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x$ (which equals $\tfrac83$).
把 $[0,2]$ 上 $f(x)=x^2$ 下的面积写成定积分。
- $n$ 条中每条的宽:$\Delta x=\dfrac{2-0}{n}=\dfrac2n$。
- 黎曼和:$\displaystyle\sum_{i=1}^{n} f(x_i)\,\Delta x$。
- 精确面积:$\displaystyle\int_0^2 x^2\,dx=\lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x$(它等于 $\tfrac83$)。
A Riemann sum in summation notation is $\sum_{i=1}^{n} f(x_i)\,\Delta x$ with $\Delta x=\frac{b-a}{n}$. Its limit as $n\to\infty$ is the definite integral $\int_a^b f(x)\,dx$ — the exact signed area, with $a,b$ the limits of integration and $f(x)$ the integrand.
求和记号下的黎曼和是 $\sum_{i=1}^{n} f(x_i)\,\Delta x$,其中 $\Delta x=\frac{b-a}{n}$。它在 $n\to\infty$ 时的极限是定积分 $\int_a^b f(x)\,dx$——精确的带符号面积,$a,b$ 是积分限,$f(x)$ 是被积函数。