Volume with Disc Method: Revolving Around the x- or y-Axis
Introduced| English | Meaning |
|---|---|
| disc/dɪsk/ | A filled circular region. |
| disc method/dɪsk ˈmeθəd/ | Finding a solid's volume by integrating the areas of filled circular cross-sections. |
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.
Each slice in the disc method is a...
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Revolving sweeps out circular discs.
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:
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$$V=\pi\int_a^b \big[R(x)\big]^2\,dx$$
- The radius $R(x)$ is the distance from the curve to the axis of revolution — here just $f(x)$.
The profile that spins into discs
y = a·√x
Revolving $y=\sqrt x$ about the $x$-axis stacks discs of radius $\sqrt x$ — volume $\pi\int R^2\,dx$.
Revolving $y=R(x)$ about the $x$-axis, the volume is...
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Each disc is $\pi R^2\,dx$.
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.
In the disc method, the radius $R$ is the ____ from the curve to the axis of revolution.
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Radius = distance to the axis.
Revolving about the $y$-axis, you integrate in $y$ using...
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Match the variable to the axis: $y$-axis → $R(y)$, integrate in $y$.
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.
Revolving $y=\sqrt x$ on $[0,4]$ about the $x$-axis ($R^2=x$) gives $V=$
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$\pi\int_0^4 x\,dx=\pi\cdot8=8\pi$.
The disc method includes a factor of $\pi$ and squares the radius.
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$V=\pi\int R^2$.
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.
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.
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.
Write from a disc volume of 8 pi
- The region under y = square root of x, from 0 to 4, is revolved about the x-axis.
- Each slice is a disc.
What is the volume?
The integral of the radius squared is 8, then multiply by pi.
Discs of radius squared x from 0 to 4 have volume what?
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pi times 8 is the volume.