AP Calculus BC / Unit 8: Applications of Integration
8.10 Volume with Disc Method: Revolving Around Other Axes
Notes
For a horizontal line $y=k$: $R=|f(x)-k|$. For a vertical line $x=h$: write $x$ in terms of $y$ and use $R=|g(y)-h|$.
Practice
The region between $y=x^2$ and $y=1$ revolved around $y=1$ on $[-1,1]$ has radius:
Distance from the line $y=1$ down to the curve.
Revolving around $y=-2$, a curve at height $f(x)\ge0$ has radius:
Distance from $-2$ up to $f(x)$.
Revolving around $x=5$, a curve $x=g(y)\le5$ has radius:
Distance from the curve to the line.
Ready for more? Take the Unit 8 test.