Approximating Areas with Riemann Sums · 用黎曼和近似面积
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| Riemann sum/ˈriːmən sʌm/ | 黎曼和 | lí màn hé |
| left/left/ | 左端 | zuǒ duān |
| right/raɪt/ | 右端 | yòu duān |
| midpoint/ˈmɪdpɔɪnt/ | 中点 | zhōng diǎn |
| trapezoid/ˈtræpɪzɔɪd/ | 梯形 | tī xíng |
| trapezoidal rule/ˈtræpɪzɔɪdl ruːl/ | 梯形法则 | tī xíng fǎ zé |
Estimate a curvy area with rectangles
- The area under a curve is rarely a neat rectangle — so approximate it with many thin rectangles.
- Slice the interval into strips, build a rectangle on each, and add up their areas.
- This sum of rectangle areas is a Riemann sum 黎曼和.
- More, thinner rectangles → a better estimate of the true area.
用矩形估计弯曲的面积
- 曲线下的面积很少是整齐的矩形——所以用许多细矩形来近似。
- 把区间切成条,在每条上建一个矩形,把它们的面积加起来。
- 这个矩形面积之和就是黎曼和。
- 矩形越多越细 → 对真实面积的估计越好。
The area we approximate · 我们近似的面积
y = x²
Rectangles or trapezoids fill the shaded area under the curve — more, thinner strips estimate it better. · 矩形或梯形填充曲线下方的阴影区域——条带越多、越薄，估计就越精确。
Using more, thinner rectangles generally improves a Riemann-sum estimate. · 通常使用更多、更薄的矩形可提高黎曼和估计的精度。
Thinner strips hug the curve better. · 更薄的条带能更好地贴合曲线。
Left, right, and midpoint rectangles
- Each rectangle's height comes from the function — but where in the strip do you measure it?
- Left 左端 sum: height from the left edge of each strip.
- Right 右端 sum: height from the right edge.
- Midpoint 中点 sum: height from the middle — usually the most accurate of the three.
左端、右端、中点矩形
- 每个矩形的高来自函数——但你在条的哪里量它?
- 左端和:高取自每条的左边缘。
- 右端和:高取自右边缘。
- 中点和:高取自中间——通常三者中最准。
For · 支持 $f(x)=x^2$ on $[0,2]$ with $2$ right rectangles (width $1$): heights $f(1)=1$, $f(2)=4$. Find the right Riemann sum. · 对于 $f(x)=x^2$ 上的 $[0,2]$，使用 $2$ 个右矩形（宽度为 $1$）：高度为 $f(1)=1$, $f(2)=4$。求右黎曼和。
$1\cdot1+4\cdot1=5$.
Same setup, but a left · 左侧 Riemann sum: heights $f(0)=0$, $f(1)=1$. Find it. · 同样的设置，但使用 左 黎曼和：高度为 $f(0)=0$, $f(1)=1$。求它。
$0\cdot1+1\cdot1=1$.
Which are standard ways to pick a rectangle's height in a Riemann sum? · 哪些是黎曼和中选取矩形高度的标准方法？
Left, right, and midpoint are the standard choices. · 左、右和中点是标准选项。
Trapezoids fit even better
- Instead of a flat-topped rectangle, connect the two edge heights with a slanted top: a trapezoid 梯形.
- The trapezoidal rule 梯形法则 averages the left and right heights on each strip.
- It hugs a smooth curve more closely than rectangles, so it's usually more accurate.
- Area of each trapezoid $=\tfrac12(\text{left height}+\text{right height})\times\text{width}$.
梯形贴合得更好
- 不用平顶矩形,而用一条斜顶连接两个边缘高度:一个梯形。
- 梯形法则在每条上取左右高度的平均。
- 它比矩形更紧贴光滑曲线,所以通常更准。
- 每个梯形的面积 $=\tfrac12(\text{left height}+\text{right height})\times\text{width}$。
The ____ rule uses slanted tops (average of edge heights) for a closer fit than rectangles. · ____ 规则使用倾斜顶边（两边高度的平均值）以实现比矩形更贴合的效果。
Each trapezoid averages the left and right heights. · 每个梯形平均左高度和右高度。
Over- or underestimate?
- On an increasing function: left sums underestimate, right sums overestimate.
- On a decreasing function: it's the reverse — left over, right under.
- The trapezoidal rule's error depends on concavity (over on concave-up, under on concave-down).
- Knowing the direction of the error lets you bound the true area.
高估还是低估?
- 对递增函数:左端和低估,右端和高估。
- 对递减函数:反过来——左高右低。
- 梯形法则的误差取决于凹凸性(上凹时高估,下凹时低估)。
- 知道误差方向,就能界定真实面积。
For an increasing function, a left Riemann sum gives an... · 对于增函数，左黎曼和给出一个...
Left heights are the smaller edge on an increasing curve → underestimate. · 左高度是增函数曲线上的较小边 → 低估。
The three rectangle rules can give quite different answers for a small number of strips — they are approximations, not the exact area. And whether a left or right sum over- or under-estimates depends on whether the function is increasing or decreasing, not on the rule alone. Always check the function's behavior.
对少数几条,三种矩形法则可能给出相当不同的答案——它们是近似,不是精确面积。而左端或右端和是高估还是低估,取决于函数递增还是递减,而非仅取决于法则。永远检查函数的行为。
Estimate the area under $f(x)=x^2$ on $[0,2]$ with $2$ right rectangles (width $1$).
- Right heights: $f(1)=1$ and $f(2)=4$.
- Right sum $=1\cdot1+4\cdot1=5$.
- (The true area is $\tfrac83\approx2.67$; since $x^2$ is increasing, the right sum overestimates — as expected.)
用 $2$ 个右端矩形(宽 $1$)估计 $[0,2]$ 上 $f(x)=x^2$ 下的面积。
- 右端高度:$f(1)=1$ 与 $f(2)=4$。
- 右端和 $=1\cdot1+4\cdot1=5$。
- (真实面积是 $\tfrac83\approx2.67$;因为 $x^2$ 递增,右端和高估——正如预期。)
A Riemann sum approximates area with rectangles: left, right, or midpoint heights. The trapezoidal rule uses slanted tops (averaging edge heights) for a closer fit. For an increasing function, left sums underestimate and right sums overestimate; more strips → a better approximation.
黎曼和用矩形近似面积:左端、右端或中点高度。梯形法则用斜顶(取边缘高度平均)贴合得更好。对递增函数,左端和低估、右端和高估;条越多 → 近似越好。