Finding the Area Between Curves Expressed as Functions of x · 求用 x 的函数表示的两曲线间的面积
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| area between two curves/ˈeərɪə bɪˈtwiːn tuː kɜːvz/ | 曲线间面积 | qū xiàn jiān miàn jī |
The area trapped between two curves
- A single integral gives the area between a curve and the $x$-axis. What about the area between two curves?
- Slice the region into thin vertical strips; each strip's height is the gap between the curves.
- Add up (integrate) those heights across the interval — that's the area between two curves 曲线间面积.
- The recipe is "top minus bottom, integrated."
夹在两条曲线之间的面积
- 单个积分给出曲线与 $x$ 轴之间的面积。那么两条曲线之间的面积呢?
- 把区域切成细的竖直条;每条的高是两曲线之间的间隙。
- 把这些高沿区间加起来(积分)——就是曲线间面积。
- 配方是"上减下,再积分"。
Top minus bottom
- If $f$ is the upper curve and $g$ the lower on $[a,b]$:
-
$$A=\int_a^b \big(f(x)-g(x)\big)\,dx$$
- Each vertical strip has height $f(x)-g(x)$ and width $dx$.
- Always subtract upper minus lower, so the height (and the area) stays positive.
上减下
- 若在 $[a,b]$ 上 $f$ 是上方曲线、$g$ 是下方:
-
$$A=\int_a^b \big(f(x)-g(x)\big)\,dx$$
- 每个竖直条高 $f(x)-g(x)$,宽 $dx$。
- 永远上减下,这样高(和面积)保持为正。
A line above a parabola · 直线在抛物线上方
y = ax² + bx + c
Between the intersection points the line sits above the parabola — the vertical gap, integrated, is the enclosed area. · 在两交点之间，直线位于抛物线上方 — 垂直间距经积分后即为所围面积。
The area between an upper curve $f$ and lower curve $g$ on $[a,b]$ is... · 上曲线 $f$ 和下曲线 $g$ 在 $[a,b]$ 之间的面积是……
Upper minus lower, integrated. · 上减下，再积分。
Find the limits from intersections
- The bounds $a$ and $b$ are usually where the curves cross — set $f(x)=g(x)$ and solve.
- Those intersection $x$-values are the left and right edges of the enclosed region.
- Between them, decide which curve is on top (test a point).
- Then integrate the top-minus-bottom difference over $[a,b]$.
由交点求积分限
- 边界 $a$ 和 $b$ 通常是两曲线相交之处——令 $f(x)=g(x)$ 求解。
- 那些交点的 $x$ 值是围成区域的左右边缘。
- 在它们之间,判断哪条曲线在上(测一个点)。
- 然后在 $[a,b]$ 上积分上减下的差。
Where do $y=x+2$ and $y=x^2$ intersect? · $y=x+2$ 和 $y=x^2$ 在哪里相交？
$x^2-x-2=0\Rightarrow x=-1,2$.
To decide which curve is on top between intersections, you... · 要决定两交点之间哪条曲线在上方，你...
Plug in a test point to see which is higher. · 代入一个测试点看哪个更高。
The limits of integration come from setting the two curves ____. · 积分限来自于令两条曲线 ____ 。
Solve $f(x)=g(x)$ for the intersection $x$-values. · 解 $f(x)=g(x)$ 以求得交点的 $x$ 值。
It works even below the axis
- Because you take upper minus lower, the formula handles regions below the $x$-axis automatically.
- The $-g(x)$ term lifts everything, so no separate "signed area" bookkeeping is needed.
- A region entirely under the axis still gives a positive area with top-minus-bottom.
- That's the advantage of the gap-height approach.
即使在轴下方也有效
- 因为你取上减下,该公式自动处理 $x$ 轴下方的区域。
- $-g(x)$ 项把一切抬起,所以不需要单独的"带符号面积"记账。
- 完全在轴下方的区域,用上减下仍给出正面积。
- 这就是间隙高度方法的优势。
The area between $y=x+2$ and $y=x^2$ is $\int_{-1}^2 (x+2-x^2)\,dx$. Compute it (a decimal). · $y=x+2$ 与 $y=x^2$ 之间的面积为 $\int_{-1}^2 (x+2-x^2)\,dx$ 。计算它（小数形式）。
The integral evaluates to $\tfrac92=4.5$. · 该积分的值为 $\tfrac92=4.5$ 。
Integrating lower minus upper gives the negative of the true area. · 对（下减上）积分会得到真实面积的相反数。
Reversing the order flips the sign. · 交换顺序会翻转符号。
Always subtract upper minus lower — if you reverse it, you get a negative of the true area. Check which curve is on top by testing an $x$-value between the intersections (they can swap, which is the next lesson). And the limits come from setting the curves equal, not from the $x$-axis.
永远上减下——若你反过来,会得到真实面积的负值。测交点之间的一个 $x$ 值来检查哪条曲线在上(它们可能交换,这是下一课)。而且积分限来自令两曲线相等,不是来自 $x$ 轴。
Find the area between $y=x+2$ and $y=x^2$.
- Intersections: $x^2=x+2\Rightarrow x^2-x-2=0\Rightarrow x=-1,2$.
- On $(-1,2)$ the line is on top (test $x=0$: line $=2>0=$ parabola).
- $A=\displaystyle\int_{-1}^{2}\big((x+2)-x^2\big)\,dx=\Big[\tfrac{x^2}{2}+2x-\tfrac{x^3}{3}\Big]_{-1}^{2}=\tfrac{9}{2}$.
求 $y=x+2$ 与 $y=x^2$ 之间的面积。
- 交点:$x^2=x+2\Rightarrow x^2-x-2=0\Rightarrow x=-1,2$。
- 在 $(-1,2)$ 上直线在上(测 $x=0$:直线 $=2>0=$ 抛物线)。
- $A=\displaystyle\int_{-1}^{2}\big((x+2)-x^2\big)\,dx=\Big[\tfrac{x^2}{2}+2x-\tfrac{x^3}{3}\Big]_{-1}^{2}=\tfrac{9}{2}$。
The area between two curves (functions of $x$) is $A=\int_a^b\big(f(x)-g(x)\big)\,dx$, integrating upper minus lower over vertical strips. Find $a,b$ where the curves intersect, decide which is on top, and integrate the gap. Top-minus-bottom keeps the area positive automatically.
曲线间面积(关于 $x$ 的函数)是 $A=\int_a^b\big(f(x)-g(x)\big)\,dx$,在竖直条上上减下积分。找两曲线相交的 $a,b$,判断哪条在上,积分间隙。上减下自动保持面积为正。