Volume with Washer Method: Revolving Around Other Axes
Introduced| English | Meaning |
|---|---|
| outer/ˈaʊtə/ | In a composition f(g(x)), the function f applied to the output of g. |
| inner/ˈɪnə/ | In a composition f(g(x)), the function g evaluated first before its output is passed to f. |
Washers around a shifted line
- The last piece: revolve a two-curve region around a line that isn't the $x$- or $y$-axis.
- Combine the two ideas you just learned: washers (a gap → a ring) and a shifted axis (adjust the radius).
- The formula stays $V=\pi\int\big(R^2-r^2\big)$; only $R$ and $r$ get re-measured to the new line.
- Get both distances right and the calculus is routine.
Both radii measured to the new axis
- For a horizontal axis $y=k$: $R=|\text{far curve}-k|$ and $r=|\text{near curve}-k|$.
- Each radius is the distance from its curve to the shifted line, not to $y=0$.
- The outer 外 radius reaches the curve farther from the axis; the inner 内 radius the nearer one.
- Then $V=\pi\int\big(R^2-r^2\big)$ as usual.
Washers about y = −1: outer and inner radii are both measured from the axis
Two curves above a shifted axis
y = ax² + bx
Revolving about $y=-1$, both radii grow by $1$ — outer $\sqrt x+1$, inner $x+1$; still $R^2-r^2$.
Revolving the region between $y=\sqrt x$ and $y=x$ about $y=-1$, the outer and inner radii are...
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Shift both radii by $+1$ down to $y=-1$.
For a shifted axis, you must re-measure ____ radii (outer and inner) to the new line.
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Shift both, not just one.
Watch which curve is now farther
- Shifting the axis can swap which curve is outer vs. inner — re-check after moving the line.
- Revolving about a line below the region: the lower curve is now the inner one (nearer the axis).
- Revolving about a line above the region: the roles flip.
- Always identify outer/inner relative to the actual axis of revolution.
After shifting the axis, you should...
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The axis position can swap outer/inner roles.
Still: square each radius first
- As with any washer, subtract $R^2-r^2$ — square the (shifted) radii before subtracting.
- Never $(R-r)^2$, and never reuse the plain-axis radii.
- Draw the two distance segments from the axis to each curve to keep them straight.
- Then integrate along the appropriate variable.
Even about a shifted axis, the washer integrand is $R^2-r^2$, not $(R-r)^2$.
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Always square each radius before subtracting.
The volume for $R=\sqrt x+1$, $r=x+1$ on $[0,1]$ about $y=-1$ is...
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$\pi\int_0^1(2\sqrt x-x-x^2)\,dx=\pi(\tfrac43-\tfrac12-\tfrac13)=\tfrac{\pi}{2}$.
The shifted-washer method combines which two ideas?
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Washer (gap → ring) plus a shifted radius.
Two combined pitfalls here: (1) shift both radii to the new axis (not just one), and (2) still use $R^2-r^2$, not $(R-r)^2$. After shifting, re-check which curve is outer — the axis's position can swap the roles. Sketch both radius segments to the actual line before squaring.
Revolve the region between $y=\sqrt x$ (top) and $y=x$ (bottom) on $[0,1]$ about the line $y=-1$.
- Shift both radii down to $y=-1$: outer $R=\sqrt x+1$, inner $r=x+1$.
- $V=\pi\displaystyle\int_0^1\Big((\sqrt x+1)^2-(x+1)^2\Big)\,dx=\pi\int_0^1\big(x+2\sqrt x - x^2 - 2x\big)\,dx$.
- $=\pi\int_0^1\big(2\sqrt x - x - x^2\big)\,dx=\pi\big(\tfrac43-\tfrac12-\tfrac13\big)=\tfrac{\pi}{2}$.
The washer method about a shifted axis: $V=\pi\int\big(R^2-r^2\big)$, but re-measure both the outer and inner radii to that line (e.g. add the shift for $y=k$ below the region). Re-check which curve is outer, and always square each radius before subtracting — never $(R-r)^2$.