Disc Method: Revolving Around x/y Axes - AP Calculus AB

1. What is the role of the function $f(x)$ in the disc method when revolving around the x-axis?

   Defines the radius of discs. Function determines how far each disc extends from the axis.

2. Which axis is used for integration when revolving around the x-axis?

   x-axis. Integration variable matches the axis of revolution.

3. Find the volume of the solid obtained by revolving $y=\frac{1}{x}$ from $x=1$ to $x=2$ around the x-axis.

   $V = \frac{\pi}{2}$. $V = \pi \int_1^2 \frac{1}{x^2} dx = \pi [-\frac{1}{x}]_1^2 = \frac{\pi}{2}$

4. Find the volume of the solid obtained by revolving $y=5$ from $x=0$ to $x=2$ around the x-axis.

   $V = 50\pi$. $V = \pi \int_0^2 25 dx = 25\pi [x]_0^2 = 50\pi$

5. Find the volume of the solid obtained by revolving $y=x^2$ from $x=0$ to $x=2$ around the x-axis.

   $V = \frac{32\pi}{5}$. $V = \pi \int_0^2 x^4 dx = \pi [\frac{x^5}{5}]_0^2 = \frac{32\pi}{5}$

6. Find the volume of the solid obtained by revolving $y=2$ from $x=0$ to $x=3$ around the x-axis.

   $V = 12\pi$. $V = \pi \int_0^3 4 dx = 4\pi [x]_0^3 = 12\pi$

7. What does $f(x)$ represent in the disc method formula $V = \int_a^b \pi [f(x)]^2 \, dx$?

   Radius of the disc. Distance from function to x-axis forms the disc radius.

8. What is the radius of each disc when revolving $x=g(y)$ around the y-axis?

   $g(y)$. Function value gives distance from y-axis to curve.

9. What is the role of the function $g(y)$ in the disc method when revolving around the y-axis?

   Defines the radius of discs. Function determines how far each disc extends from the axis.

10. State the geometric shape formed by the cross-section in the disc method.

    Circle. Cross-sections perpendicular to the axis are circular discs.

11. Find the volume of the solid obtained by revolving $x=y^2$ from $y=0$ to $y=2$ around the y-axis.

    $V = \frac{32\pi}{5}$. $V = \pi \int_0^2 y^4 dy = \pi [\frac{y^5}{5}]_0^2 = \frac{32\pi}{5}$

12. What is the integral setup for a disc method problem revolving $y=x^3$ from $x=0$ to $x=1$?

    $V = \int_0^1 \pi (x^3)^2 \, dx$. Disc method setup with radius $x^3$ squared in integrand.

13. In the disc method, what shape is formed when revolving a function around an axis?

    Solid of revolution. Revolution creates a 3D solid from the 2D region.

14. What is the cross-sectional area of a disc with radius $r$?

    $\pi r^2$. Standard formula for area of a circle.

15. Identify the variable of integration when revolving around the y-axis using the disc method.

    $y$. Integration follows the axis of revolution.

16. Identify the variable of integration when revolving around the x-axis using the disc method.

    $x$. Integration follows the axis of revolution.

17. Identify the axis of symmetry for the volume of revolution problem using the disc method.

    The axis of revolution. Solid rotates around this line creating circular symmetry.

18. Which axis is used for integration when revolving around the y-axis?

    y-axis. Integration variable matches the axis of revolution.

19. What is the radius of each disc when revolving $y=f(x)$ around the x-axis?

    $f(x)$. Function value gives distance from x-axis to curve.

20. Find the volume of the solid obtained by revolving $x=\frac{1}{y}$ from $y=1$ to $y=2$ around the y-axis.

    $V = \frac{\pi}{2}$. $V = \pi \int_1^2 \frac{1}{y^2} dy = \pi [-\frac{1}{y}]_1^2 = \frac{\pi}{2}$

21. What does $g(y)$ represent in the disc method formula $V = \int_c^d \pi [g(y)]^2 \, dy$?

    Radius of the disc. Distance from function to y-axis forms the disc radius.

22. What is the volume formula for a solid of revolution around the y-axis using the disc method?

    $V = \int_c^d \pi [g(y)]^2 \, dy$. Formula integrates $\pi$ times the radius squared along the y-axis.

23. What is the integral setup for a disc method problem revolving $y=x^2$ from $x=0$ to $x=1$?

    $V = \int_0^1 \pi (x^2)^2 \, dx$. Disc method setup with radius $x^2$ squared in integrand.

24. What is the formula for the volume of a solid of revolution using the disc method around the x-axis?

    $V = \int_a^b \pi [f(x)]^2 \, dx$. Formula integrates $\pi$ times the radius squared over the interval.

25. Given $y=f(x)$, identify the limits of integration when revolving from $x=a$ to $x=b$ around the x-axis.

    $a$ to $b$. Limits match the given x-interval for the region.

26. Find the volume of the solid obtained by revolving $y=\frac{1}{x}$ from $x=1$ to $x=2$ around the x-axis.

    $V = \frac{\pi}{2}$. $V = \pi \int_1^2 \frac{1}{x^2} dx = \pi [-\frac{1}{x}]_1^2 = \frac{\pi}{2}$

27. Find the volume of the solid obtained by revolving $x=2$ from $y=0$ to $y=4$ around the y-axis.

    $V = 16\pi$. $V = \pi \int_0^4 4 dy = 4\pi [y]_0^4 = 16\pi$

28. Find the volume of the solid obtained by revolving $y=3x$ from $x=0$ to $x=1$ around the x-axis.

    $V = 3\pi$. $V = \pi \int_0^1 9x^2 dx = 3\pi [x^3]_0^1 = 3\pi$

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