Finding the Area Between Curves Expressed as Functions of y · 求用 y 的函数表示的两曲线间的面积
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| horizontal slice/ˌhɒrɪˈzɒntl slaɪs/ | 水平切片 | shuǐ píng qiē piàn |
Slice sideways instead of up-and-down
- Some regions are awkward with vertical strips — a sideways-opening parabola, say.
- The fix: slice into horizontal strips and integrate with respect to $y$ instead of $x$.
- Everything from the last lesson turns 90°: curves become functions of $y$, and you subtract right minus left.
- Same idea, rotated: add up horizontal gap-widths.
横着切,而不是上下切
- 有些区域用竖直条不好处理——比如一个向侧面开口的抛物线。
- 办法:切成水平条,改为关于 $y$ 积分而非 $x$。
- 上一课的一切转 90°:曲线变成 $y$ 的函数,你用右减左。
- 同样的想法,旋转过来:把水平间隙宽度加起来。
Right minus left
- Write the boundaries as functions of $y$: $x=f(y)$ (rightmost) and $x=g(y)$ (leftmost).
-
$$A=\int_c^d \big(f(y)-g(y)\big)\,dy$$
- Each horizontal slice 水平切片 has width $f(y)-g(y)$ and height $dy$.
- Subtract rightmost minus leftmost so the width stays positive.
右减左
- 把边界写成 $y$ 的函数:$x=f(y)$(最右)和 $x=g(y)$(最左)。
-
$$A=\int_c^d \big(f(y)-g(y)\big)\,dy$$
- 每个水平切片宽 $f(y)-g(y)$,高 $dy$。
- 最右减最左,这样宽度保持为正。
A sideways-opening parabola · 开口侧向的抛物线
y = ax² + bx + c
When curves open sideways, horizontal slices (right minus left, integrated in $y$) are the clean choice. · 当曲线开口侧向时，水平切片（右减左，对 $y$ 积分）是更简洁的选择。
With horizontal slices, the area is $\int_c^d(\ ?\ )\,dy$ where... · 使用水平切片时，面积为 $\int_c^d(\ ?\ )\,dy$ ，其中...
Horizontal strips: right curve minus left curve. · 水平条带：右曲线减左曲线。
Bounds are now $y$-values
- The limits $c$ and $d$ are the lowest and highest $y$ of the region — often the curves' intersection $y$-values.
- Solve $f(y)=g(y)$ (in terms of $y$) for those bounds.
- Between them, test which curve is farther right.
- Then integrate the right-minus-left width from $c$ to $d$.
现在边界是 $y$ 值
- 边界 $c$ 和 $d$ 是区域的最低和最高 $y$——常是曲线交点的 $y$ 值。
- 解 $f(y)=g(y)$(关于 $y$)求这些边界。
- 在它们之间,测哪条曲线更靠右。
- 然后从 $c$ 到 $d$ 积分右减左的宽度。
For a $dy$ integral, the limits of integration are... · 对于关于 $dy$ 的积分，积分限是...
Integrating in $y$ means $y$-limits. · 对 $y$ 积分意味着 $y$ 的积分限。
Where do $x=y^2$ and $x=y+2$ intersect (in $y$)? · $x=y^2$ 和 $x=y+2$ 在哪里相交（在 $y$ 中）？
$y^2-y-2=0\Rightarrow y=-1,2$.
When to choose $dy$ over $dx$
- Prefer horizontal slices when the curves are naturally $x=$(function of $y$), or when vertical slices would need splitting.
- A region bounded on the left and right by different curves (but top/bottom by single points) is a $dy$ situation.
- If the "upper" curve would change midway with vertical slices, horizontal slices may avoid the split.
- Pick whichever orientation makes each strip span one clean curve to one clean curve.
何时选 $dy$ 而非 $dx$
- 当曲线自然是 $x=$($y$ 的函数),或竖直条会需要分段时,优先用水平切片。
- 一个左右由不同曲线、上下由单点界定的区域,是一个 $dy$ 情形。
- 若"上方"曲线用竖直条会中途改变,水平切片也许能避免分段。
- 选让每条从一条干净曲线到一条干净曲线的方向。
The area bounded by $x=y^2$ and $x=y+2$ is $\int_{-1}^2 (y+2-y^2)\,dy$. Compute it (a decimal). · $x=y^2$ 与 $x=y+2$ 围成的面积为 $\int_{-1}^2 (y+2-y^2)\,dy$ 。计算它（小数形式）。
It evaluates to $\tfrac92=4.5$. · 其值为 $\tfrac92=4.5$ 。
For a horizontal-slice integral, you should write each boundary as $x$ in terms of $y$. · 对于水平切片积分，应将每条边界写成 $x$ 关于 $y$ 的表达式。
The strips run horizontally, so express $x$ as a function of $y$. · 因为条带是水平的，所以要将 $x$ 表示为 $y$ 的函数。
Horizontal slices are preferable when... · 水平切片在以下情况下更优：...
Sideways curves suit $dy$ integration. · 侧向曲线适合对 $dy$ 积分。
With horizontal slices you integrate right minus left in terms of $y$, and the limits are $y$-values (not $x$-values). Mixing them up — using $x$-limits with a $dy$ integral, or subtracting top-minus-bottom sideways — gives a wrong answer. Rewrite each boundary curve as $x$ in terms of $y$ first.
用水平切片时,你关于 $y$ 积分右减左,边界是 $y$ 值(不是 $x$ 值)。把它们搞混——对 $dy$ 积分用 $x$ 边界,或横着上减下——会给出错误答案。先把每条边界曲线改写成 $x$ 关于 $y$ 的形式。
Find the area bounded by $x=y^2$ and $x=y+2$.
- Intersections (in $y$): $y^2=y+2\Rightarrow y^2-y-2=0\Rightarrow y=-1,2$.
- On $(-1,2)$ the line $x=y+2$ is farther right (test $y=0$: $2>0$).
- $A=\displaystyle\int_{-1}^{2}\big((y+2)-y^2\big)\,dy=\tfrac{9}{2}$.
求由 $x=y^2$ 与 $x=y+2$ 围成的面积。
- 交点(关于 $y$):$y^2=y+2\Rightarrow y^2-y-2=0\Rightarrow y=-1,2$。
- 在 $(-1,2)$ 上,直线 $x=y+2$ 更靠右(测 $y=0$:$2>0$)。
- $A=\displaystyle\int_{-1}^{2}\big((y+2)-y^2\big)\,dy=\tfrac{9}{2}$。
For curves given as functions of $y$, use horizontal slices: $A=\int_c^d\big(f(y)-g(y)\big)\,dy$, integrating rightmost minus leftmost with $y$-limits from the intersections. Choose $dy$ when it lets each strip run cleanly from one curve to another without splitting.
对以 $y$ 为自变量的曲线,用水平切片:$A=\int_c^d\big(f(y)-g(y)\big)\,dy$,以交点的 $y$ 边界最右减最左积分。当它能让每条从一曲线干净地到另一曲线而无需分段时,选 $dy$。