Disc Method: Revolving Around Other Axes - AP Calculus BC

1. Describe how to use the disc method when revolving around a non-axis line.

   Adjust radius as $|f(x) - \text{line}|$. The distance from curve to the line becomes the radius.

2. What is the effect of changing the axis of rotation to $y = c$?

   Radius becomes $|f(x) - c|$. Shift from x-axis changes radius to distance from line.

3. What is the role of the function $g(y)$ in the disc method when using the y-axis?

   It defines the radius for integration. Function $g(y)$ gives horizontal distance from y-axis.

4. Identify the expression for the radius in the disc method revolving around the y-axis.

   Radius = $g(y)$. The function $g(y)$ gives distance from y-axis to curve.

5. How do you adjust the disc method formula for revolving around the y-axis?

   Use $x = f(y)$ instead of $y = f(x)$. Switch to integrating with respect to $y$ and use horizontal cross-sections.

6. How is the disc method implemented for a solid of revolution about $y = b$?

   Adjust radius: $|g(y) - b|$. Distance from curve to horizontal line $y=b$ becomes radius.

7. What adjustment is made to the disc method for a non-axis horizontal line?

   Adjust radius: $|f(x) - \text{line}|$. Distance from function to horizontal line becomes radius.

8. What adjustment is made to the disc method for a non-axis vertical line?

   Adjust radius: $|g(y) - \text{line}|$. Distance from function to vertical line becomes radius.

9. How do you determine the limits of integration for revolving around the x-axis?

   Use the interval $[a, b]$ of $x$. Integration bounds match the domain of the function.

10. What is the role of the limits of integration in the disc method?

    They define the bounds of integration. Limits specify the interval over which to integrate.

11. What change is made to the disc method when revolving around a horizontal line $y = c$?

    Use $|f(x) - c|$ as radius. Distance from curve to line $y=c$ becomes the new radius.

12. How do you determine the limits of integration for revolving around the y-axis?

    Use the interval $[c, d]$ of $y$. Integration bounds match the range of the function.

13. What is the effect of changing the axis of rotation to $x = c$?

    Radius becomes $|g(y) - c|$. Shift from y-axis changes radius to distance from line.

14. How does the function $f(x)$ affect the volume when using the disc method?

    It determines the radius. Function values determine cross-sectional radii at each point.

15. What change is made to the disc method when revolving around a vertical line $x = c$?

    Use $|g(y) - c|$ as radius. Distance from curve to line $x=c$ becomes the new radius.

16. Identify the expression for the radius in the disc method revolving around the x-axis.

    Radius = $f(x)$. The function value gives the distance from x-axis to curve.

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