Finding the Area Between Curves That Intersect at More Than Two Points · 求相交点多于两个的曲线间的面积
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| subintervals/ˌsʌbɪnˈtɜːvlz/ | 子区间 | zi qū jiān |
When the curves keep crossing
- Sometimes two curves cross more than twice, and the "top" curve swaps along the way.
- A single top-minus-bottom integral would then get the sign wrong on part of the region.
- The fix: split the region at each crossing and integrate each piece separately.
- Add the pieces' (positive) areas to get the total.
当曲线反复相交
- 有时两条曲线相交多于两次,"上方"曲线沿途交换。
- 那么单个上减下的积分会在区域的一部分上弄错符号。
- 办法:在每个交点处分段,分别积分每一块。
- 把各块的(正)面积相加得到总量。
Find every intersection first
- Solve $f(x)=g(x)$ completely — find all the crossing points on the interval, not just the outer two.
- These crossings are the boundaries of the subintervals 子区间 where the ordering is fixed.
- Order them left to right along the $x$-axis.
- Each adjacent pair of crossings frames one piece of the region.
先找出每个交点
- 完整地解 $f(x)=g(x)$——找区间上所有交点,不只是最外两个。
- 这些交点是次序固定的子区间的边界。
- 沿 $x$ 轴从左到右排列它们。
- 每对相邻交点框出区域的一块。
A cubic crossing a line 3 times · 三次曲线与直线相交 3 次
y = ax³ + bx
$y=x^3$ and $y=x$ cross at $-1,0,1$; the top curve switches at $0$, so the area needs two pieces. · $y=x^3$ 和 $y=x$ 在 $-1,0,1$ 处相交；上曲线在 $0$ 处切换，因此面积需要分为两部分。
Where do $y=x^3$ and $y=x$ cross? · $y=x^3$ 和 $y=x$ 在哪里相交？
$x^3=x\Rightarrow x(x-1)(x+1)=0$.
The intersection points divide the region into ____ where the ordering is fixed. · 交点将区域划分为____，其中顺序是固定的。
On each subinterval one curve is consistently on top. · 在每个子区间上，一条曲线始终位于上方。
Split where top and bottom switch
- On each subinterval, one curve is consistently on top — test a point to see which.
- Set up a separate integral of (upper $-$ lower) for each subinterval, using the correct top curve there.
- Because the top curve switches between pieces, the integrands differ from piece to piece.
- The full area is the sum of these separate integrals.
在上下互换处分段
- 在每个子区间上,一条曲线始终在上——测一个点看是哪条。
- 为每个子区间建立一个单独的(上 $-$ 下)积分,用那里正确的上方曲线。
- 因为上方曲线在各块之间互换,被积函数逐块不同。
- 全部面积是这些单独积分之和。
You split the region at each intersection because... · 你在每个交点处分割区域是因为...
Where curves swap, part of the region would count negative. · 当曲线互换时，部分区域会被计为负值。
Using one integral of $f-g$ across all crossings (curves swapping) gives... · 在整个交叉区间（曲线互换）上使用一个 $f-g$ 的积分会得到...
Sign-cancellation understates the true area. · 符号相减低估了真实面积。
Add the absolute areas
- Each subinterval integral, done with the correct top-minus-bottom, is positive.
- Sum them: $A=\displaystyle\sum \int_{\text{piece}} (\text{upper}-\text{lower})\,dx$.
- Equivalently, $A=\displaystyle\int_a^b |f(x)-g(x)|\,dx$ — the absolute gap, integrated.
- Both viewpoints give the same total geometric area.
相加绝对面积
- 每个子区间用正确的上减下算出的积分都是正的。
- 相加:$A=\displaystyle\sum \int_{\text{piece}} (\text{upper}-\text{lower})\,dx$。
- 等价地,$A=\displaystyle\int_a^b |f(x)-g(x)|\,dx$——绝对间隙的积分。
- 两个视角给出相同的总几何面积。
For · 支持 $y=x^3$ and $y=x$ on $[-1,1]$, each piece has area $\tfrac14$. Find the total area. · 对于 $y=x^3$ 和 $y=x$ 在 $[-1,1]$ 上，每部分的面积为 $\tfrac14$ 。求总面积。
$\tfrac14+\tfrac14=\tfrac12$.
Integrating $|f(x)-g(x)|$ over the whole interval gives the same total area as splitting. · 对整个区间积分 $|f(x)-g(x)|$ 得到的总面积与分段计算相同。
The absolute value handles the switching automatically. · 绝对值自动处理了切换问题。
Do not use one integral of $f-g$ across all crossings — where the curves swap, part of the region contributes a negative value and cancels real area. You must split at every intersection and take upper-minus-lower correctly on each piece (or integrate $|f-g|$). Find all the crossings, not just the endpoints.
不要跨所有交点用一个 $f-g$ 的积分——在曲线互换处,区域的一部分贡献负值并抵消真实面积。你必须在每个交点分段,在每块上正确地取上减下(或积分 $|f-g|$)。找所有交点,不只是端点。
Area between $y=x^3$ and $y=x$ on $[-1,1]$.
- Crossings: $x^3=x\Rightarrow x(x^2-1)=0\Rightarrow x=-1,0,1$ — three points.
- On $(-1,0)$: $x^3>x$ (test $-0.5$). On $(0,1)$: $x>x^3$ (test $0.5$). The top curve switches at $0$.
- $A=\displaystyle\int_{-1}^{0}(x^3-x)\,dx+\int_{0}^{1}(x-x^3)\,dx=\tfrac14+\tfrac14=\tfrac12$.
$[-1,1]$ 上 $y=x^3$ 与 $y=x$ 之间的面积。
- 交点:$x^3=x\Rightarrow x(x^2-1)=0\Rightarrow x=-1,0,1$——三个点。
- 在 $(-1,0)$:$x^3>x$(测 $-0.5$)。在 $(0,1)$:$x>x^3$(测 $0.5$)。上方曲线在 $0$ 处互换。
- $A=\displaystyle\int_{-1}^{0}(x^3-x)\,dx+\int_{0}^{1}(x-x^3)\,dx=\tfrac14+\tfrac14=\tfrac12$。
When curves intersect more than twice, find all intersection points, split the region into subintervals, and integrate upper minus lower on each (the top curve switches between them). Sum the pieces — equivalently, integrate $|f(x)-g(x)|$. One straight integral of $f-g$ would wrongly cancel area.
当曲线相交多于两次,找所有交点,把区域分成子区间,在每个上积分上减下(上方曲线在它们之间互换)。把各块相加——等价于积分 $|f(x)-g(x)|$。跨所有交点的一个 $f-g$ 积分会错误地抵消面积。