Volume with Disc Method: Revolving Around the x- or y-Axis · 圆盘法体积：绕 x 轴或 y 轴旋转
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| disc/dɪsk/ | 圆盘 | yuán pán |
| disc method/dɪsk ˈmeθəd/ | 圆盘法 | yuán pán fǎ |
Spin a region into a solid
- Take a region, spin it around an axis, and it sweeps out a solid of revolution.
- Slice perpendicular to the axis and each slice is a thin disc 圆盘 (a coin).
- Add up the disc volumes: the disc method 圆盘法.
- It's the cross-section formula with circular slices.
把区域旋转成立体
- 取一个区域,绕一条轴旋转,它就扫出一个旋转体。
- 垂直于轴切片,每片是一个薄圆盘(一枚硬币)。
- 把圆盘体积加起来:圆盘法。
- 这是横截面公式配上圆形切片。
Each slice in the disc method is a... · 圆盘法中的每个切片是一个...
Revolving sweeps out circular discs. · 旋转扫出圆形圆盘。
The disc formula
- Each disc is a circle of radius $R$ and thickness $dx$, so its volume is $\pi R^2\,dx$.
- If the region between $y=R(x)$ and the axis is revolved about the $x$-axis:
-
$$V=\pi\int_a^b \big[R(x)\big]^2\,dx$$
- The radius $R(x)$ is the distance from the curve to the axis of revolution — here just $f(x)$.
圆盘公式
- 每个圆盘是半径 $R$、厚 $dx$ 的圆,所以它的体积是 $\pi R^2\,dx$。
- 若 $y=R(x)$ 与轴之间的区域绕 $x$ 轴旋转:
-
$$V=\pi\int_a^b \big[R(x)\big]^2\,dx$$
- 半径 $R(x)$ 是从曲线到旋转轴的距离——这里就是 $f(x)$。
The profile that spins into discs · 旋转形成圆盘的轮廓
y = a·√x
Revolving $y=\sqrt x$ about the $x$-axis stacks discs of radius $\sqrt x$ — volume $\pi\int R^2\,dx$. · 绕 $y=\sqrt x$ 轴旋转的 $x$ 堆积出半径为 $\sqrt x$ 的圆盘——体积 $\pi\int R^2\,dx$。
Revolving $y=R(x)$ about the $x$-axis, the volume is... · 绕$y=R(x)$旋转$x$轴，体积为...
Each disc is $\pi R^2\,dx$. · 每个圆盘是$\pi R^2\,dx$。
Revolving about the y-axis
- Spin about the $y$-axis instead? Slice horizontally and write the radius in terms of $y$.
-
$$V=\pi\int_c^d \big[R(y)\big]^2\,dy$$
- Now $R(y)$ is the horizontal distance from the curve $x=g(y)$ to the $y$-axis.
- Same formula, integrated in $y$, with the radius measured sideways.
绕 y 轴旋转
- 改绕 $y$ 轴旋转?水平切片,把半径写成 $y$ 的函数。
-
$$V=\pi\int_c^d \big[R(y)\big]^2\,dy$$
- 现在 $R(y)$ 是从曲线 $x=g(y)$ 到 $y$ 轴的水平距离。
- 同样的公式,关于 $y$ 积分,半径横向测量。
In the disc method, the radius $R$ is the ____ from the curve to the axis of revolution. · 在圆盘法中，半径$R$是曲线到旋转轴的____距离。
Radius = distance to the axis. · 半径 = 到轴的距离。
Revolving about the $y$-axis, you integrate in $y$ using... · 绕$y$轴旋转时，你对$y$进行积分，使用...
Match the variable to the axis: $y$-axis → $R(y)$, integrate in $y$. · 变量与轴匹配：$y$轴 → $R(y)$，对$y$积分。
The radius is a distance
- The key skill: correctly identify $R$ as the distance from the curve to the axis.
- Revolving about the $x$-axis, a curve $y=f(x)$ gives radius $R=f(x)$ (its height).
- Revolving about the $y$-axis, a curve $x=g(y)$ gives radius $R=g(y)$.
- Square the radius, multiply by $\pi$, and integrate along the axis.
半径是一段距离
- 关键技能:正确认出 $R$ 是从曲线到轴的距离。
- 绕 $x$ 轴旋转,曲线 $y=f(x)$ 给出半径 $R=f(x)$(它的高)。
- 绕 $y$ 轴旋转,曲线 $x=g(y)$ 给出半径 $R=g(y)$。
- 把半径平方,乘以 $\pi$,沿轴积分。
Revolving $y=\sqrt x$ on $[0,4]$ about the $x$-axis ($R^2=x$) gives $V=$ · $y=\sqrt x$绕$[0,4]$的$x$轴（$R^2=x$）旋转得到$V=$
$\pi\int_0^4 x\,dx=\pi\cdot8=8\pi$.
The disc method includes a factor of $\pi$ and squares the radius. · 圆盘法包含一个$\pi$因子并对半径进行平方。
$V=\pi\int R^2$.
Don't forget the $\pi$ and the squaring — the disc method is $\pi\int R^2$, not $\int R$. And match the integration variable to the axis: revolve about the $x$-axis → integrate in $x$ with $R(x)$; about the $y$-axis → integrate in $y$ with $R(y)$. Mixing them is a common slip.
别忘了 $\pi$ 和平方——圆盘法是 $\pi\int R^2$,不是 $\int R$。并把积分变量匹配到轴:绕 $x$ 轴旋转 → 关于 $x$ 积分,用 $R(x)$;绕 $y$ 轴 → 关于 $y$ 积分,用 $R(y)$。搞混是常见失误。
Revolve the region under $y=\sqrt x$ on $[0,4]$ about the $x$-axis.
- Radius $R(x)=\sqrt x$, so $R^2=x$.
- $V=\pi\displaystyle\int_0^4 x\,dx=\pi\Big[\tfrac{x^2}{2}\Big]_0^4=8\pi$.
- Each disc is a circle of radius $\sqrt x$ stacked along the $x$-axis.
把 $[0,4]$ 上 $y=\sqrt x$ 下方的区域绕 $x$ 轴旋转。
- 半径 $R(x)=\sqrt x$,所以 $R^2=x$。
- $V=\pi\displaystyle\int_0^4 x\,dx=\pi\Big[\tfrac{x^2}{2}\Big]_0^4=8\pi$。
- 每个圆盘是半径 $\sqrt x$ 的圆,沿 $x$ 轴叠起。
The disc method revolves a region against an axis into circular slices: $V=\pi\int_a^b[R(x)]^2\,dx$ about the $x$-axis, or $\pi\int_c^d[R(y)]^2\,dy$ about the $y$-axis. The radius $R$ is the distance from the curve to the axis. Never drop the $\pi$ or the square.
圆盘法把区域贴着一条轴旋转成圆形切片:绕 $x$ 轴 $V=\pi\int_a^b[R(x)]^2\,dx$,绕 $y$ 轴 $\pi\int_c^d[R(y)]^2\,dy$。半径 $R$ 是从曲线到轴的距离。永远别丢 $\pi$ 或平方。