Disc Method: Revolving Around Other Axes - AP Calculus AB

1. What does $b-a$ represent in the formula $V = \pi \\int_{a}^{b} [f(x)]^2 \, dx$?

   The interval length along the x-axis. This represents the width of the integration domain.

2. What is the purpose of squaring the function in the disc method formula?

   Squaring gives the area of the disc cross-section. The squared radius gives the circular cross-sectional area.

3. Determine the axis of revolution for $V = \pi \\int_{a}^{b} [f(x)]^2 \, dx$.

   The axis of revolution is the x-axis. Integration with respect to $x$ indicates x-axis revolution.

4. What is the effect of increasing the outer radius on the volume of the solid?

   Increases the volume. Larger radius creates proportionally larger volume.

5. Identify the correct limits for $V = \pi \\int_{a}^{b} [f(x)]^2 \, dx$ if revolving around x-axis from $x = 2$ to $x = 5$.

   The limits are $2$ to $5$. Integration limits correspond to the revolution interval.

6. Find the volume using the disc method for $f(x) = 3$ from $x = 0$ to $x = 2$, revolving around the x-axis.

   $V = 18\\pi$. Using $V = \pi \int_0^2 (3)^2 dx = \pi \cdot 9 \cdot 2 = 18\pi$.

7. What is the role of $\\pi$ in the disc method formula?

   $\\pi$ scales the area to the volume of the disc. Converts the squared radius to circular area.

8. What is the effect of changing the axis of revolution on the disc method?

   It alters the radius function and limits. Different axes require different radius functions and variables.

9. State the formula for volume revolving $y = g(x)$ around the x-axis using the disc method.

   $V = \pi \int_{a}^{b} [g(x)]^2 \, dx$. General form where $g(x)$ is the radius function.

10. Identify the function for $V = \pi \\int_{1}^{4} [h(x)]^2 \, dx$.

    The function is $h(x)$. The function inside the brackets is the radius function.

11. State the formula for the volume of a solid with a hole using the disc method.

    $V = \pi \int_{a}^{b} ([R(x)]^2 - [r(x)]^2) \, dx$. Washer method subtracts inner from outer disc areas.

12. Identify the formula for volume revolving $x = h(y)$ around the y-axis.

    $V = \pi \int_{c}^{d} [h(y)]^2 \, dy$. Standard formula for y-axis revolution with $h(y)$ as radius.

13. Find the volume using the disc method for $f(x) = x$ from $x = 0$ to $x = 1$, around the x-axis.

    $V = \frac{\pi}{3}$. Using $V = \pi \int_0^1 x^2 dx = \frac{\pi}{3}$.

14. State the formula for volume using the disc method around the x-axis.

    $V = \pi \\int_{a}^{b} [f(x)]^2 \, dx$. Standard disc method formula where $f(x)$ is the radius function.

15. Identify the correct function for $V = \pi \\int_{0}^{2} [2y]^2 \, dy$ around the y-axis.

    The function is $2y$. The radius function is extracted from the integrand.

16. State the integration variable for revolving a function around the y-axis.

    The integration variable is $y$. Y-axis revolution requires integration with respect to $y$.

17. State the role of the limits of integration in the disc method formula.

    Determine the interval of revolution. Integration bounds define the region of revolution.

18. State the limits of integration for rotating $f(x)$ from $x=a$ to $x=b$ around the x-axis.

    The limits are $a$ to $b$. Integration bounds match the domain of revolution.

19. Identify the effect of a larger integration interval on volume using the disc method.

    Increases volume. Wider intervals encompass more cross-sectional discs.

20. What is the role of $\pi$ in the disc method formula?

    $\pi$ scales the area to the volume of the disc. Converts the squared radius to circular area.

21. Identify the outer radius for $V = \pi \\int_{0}^{3} [\sqrt{x}]^2 \, dx$ around the x-axis.

    The outer radius is $\sqrt{x}$. The function inside the brackets is the radius.

22. What is the result of integrating a negative function in the disc method?

    Produces incorrect volume; function must be non-negative. Squaring negative values gives positive areas erroneously.

23. State the formula for the disc area in the disc method.

    Disc area is $A = \pi [r(x)]^2$. Standard area formula for a circle with radius $r(x)$.

24. Find the volume using the disc method for $f(x) = x^2$ from $x = 1$ to $x = 2$, around the x-axis.

    $V = \frac{31\pi}{5}$. Using $V = \pi \int_1^2 (x^2)^2 dx = \pi \int_1^2 x^4 dx = \frac{31\pi}{5}$

25. Find the volume using the disc method for $f(x) = 2$ from $x = 0$ to $x = 3$, revolving around the x-axis.

    $V = 12\pi$. Using $V = \pi \int_0^3 (2)^2 dx = \pi \cdot 4 \cdot 3 = 12\pi$.

26. What is the inner radius in the disc method when there's no hole?

    The inner radius is zero. No cavity means the solid extends to the axis of revolution.

27. What is the outer radius in the disc method when revolving around the y-axis?

    The outer radius is $R = h(y)$, the distance from the axis to the outer curve. Distance from y-axis to the curve defines the radius.

28. Find the volume using the disc method for $g(y) = y^2$ from $y = 0$ to $y = 2$, revolving around the y-axis.

    $V = \frac{32\pi}{5}$. Using $V = \pi \int_0^2 (y^2)^2 dy = \pi \int_0^2 y^4 dy = \frac{32\pi}{5}$.

29. State the volume formula using the disc method around the y-axis.

    $V = \pi \int_{c}^{d} [g(y)]^2 \, dy$. Standard formula for revolution around the y-axis.

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