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

1. Define the disc method in calculus.

   A method to find volume by revolving a region around an axis. Creates circular cross-sections perpendicular to the axis.

2. What is the integral limit when revolving region from $y = 0$ to $y = 4$ around y-axis?

   $0$ to $4$. Limits match the given y-interval bounds.

3. Identify the axis of revolution for the formula: $V = \pi \int_a^b [f(y)]^2 \, dy$.

   y-axis. Variable $y$ in the integral indicates y-axis revolution.

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

   $g(y)$. Distance from y-axis to the curve is $g(y)$.

5. 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 for revolving around the x-axis.

6. What type of integral is used in the disc method?

   Definite integral. Volume calculations require definite integration.

7. Calculate the volume of the solid formed by revolving $y = \cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ around the x-axis.

   $\frac{\pi^2}{4}$. $\pi \int_0^{\pi/2} \cos^2(x) dx$ using power reduction.

8. What does the variable $c$ represent in the formula $V = \pi \int_c^d [f(y)]^2 \, dy$?

   Lower bound of integration along y-axis. Starting point of integration interval for y-axis.

9. Identify the function to be squared in the integral for the disc method around the y-axis.

   $f(y)$. For y-axis revolution, the radius function is $f(y)$.

10. Identify the axis of revolution for the formula: $V = \pi \int_c^d [g(y)]^2 \, dy$.

    y-axis. Variable $y$ in the integral indicates y-axis revolution.

11. What represents the height of the disc in the disc method?

    Infinitesimal thickness $dx$ or $dy$. Represents the width of each infinitesimal disc slice.

12. Calculate the volume of the solid formed by revolving $y = \cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ around the x-axis.

    $\frac{\pi^2}{4}$. $\pi \int_0^{\pi/2} \cos^2(x) dx$ using power reduction.

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

    Upper bound of integration along x-axis. Ending point of integration interval for x-axis.

14. Determine the volume of the solid formed by revolving $y = x^2$ from $x = 0$ to $x = 1$ around the x-axis.

    $\frac{\pi}{5}$. $\pi \int_0^1 x^4 dx = \pi[\frac{x^5}{5}]_0^1 = \frac{\pi}{5}$.

15. What is the integral limit when revolving region from $x = 1$ to $x = 3$ around x-axis?

    $1$ to $3$. Limits match the given x-interval bounds.

16. What is the result of integrating $\pi [g(y)]^2$ with respect to $y$ from $c$ to $d$?

    Volume of solid of revolution. Disc method integration gives volume of revolution.

17. Calculate the volume of the solid formed by revolving $y = \frac{1}{2}x$ from $x = 0$ to $x = 3$ around the x-axis.

    $\frac{9\pi}{8}$. $\pi \int_0^3 (\frac{x}{2})^2 dx = \pi \int_0^3 \frac{x^2}{4} dx$.

18. Calculate the volume of the solid formed by revolving $y = 2x$ from $x = 0$ to $x = 3$ around the x-axis.

    $36\pi$. $\pi \int_0^3 (2x)^2 dx = \pi \int_0^3 4x^2 dx = 36\pi$.

19. When revolving around the y-axis, what represents the height of the disc?

    Infinitesimal thickness $dy$. Width of each disc slice when integrating along y-axis.

20. What is the volume of the solid formed by revolving $x = 3$ from $y = 0$ to $y = 2$ around the y-axis?

    $18\pi$. $\pi \int_0^2 3^2 dy = \pi \int_0^2 9 dy = 18\pi$.

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

    Lower bound of integration along y-axis. Starting point of integration interval for y-axis.

22. Identify the volume formula for the solid formed by revolving $y = x$ from $x = 0$ to $x = 2$ around x-axis.

    $V = \pi \int_0^2 x^2 \, dx$. Radius is $f(x) = x$, so we integrate $\pi x^2$.

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

    $f(x)$. Distance from x-axis to the curve is $f(x)$.

24. Identify the volume formula for the solid formed by revolving $x = y^2$ from $y = 0$ to $y = 1$ around y-axis.

    $V = \pi \int_0^1 y^4 \, dy$. Radius is $f(y) = y^2$, so we integrate $\pi y^4$.

25. What does the variable $d$ represent in the formula $V = \pi \int_c^d [f(y)]^2 \, dy$?

    Upper bound of integration along y-axis. Ending point of integration interval for y-axis.

26. What is the result of integrating $\pi [f(x)]^2$ with respect to $x$ from $a$ to $b$?

    Volume of solid of revolution. Disc method integration gives volume of revolution.

27. Identify the axis of revolution for the formula: $V = \pi \int_c^d [g(y)]^2 \, dy$.

    y-axis. Variable $y$ in the integral indicates y-axis revolution.

28. Determine the volume generated by revolving $y = 1 - x^2$ from $x = -1$ to $x = 1$ around the x-axis.

    $\frac{16\pi}{15}$. $\pi \int_{-1}^1 (1-x^2)^2 dx$ evaluates to this value.

29. Identify the function to be squared in the integral for the disc method around the x-axis.

    $f(x)$. For x-axis revolution, the radius function is $f(x)$.

30. Calculate the volume of the solid formed by revolving $y = \sin(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ around the x-axis.

    $\frac{\pi^2}{4}$. $\pi \int_0^{\pi/2} \sin^2(x) dx$ using power reduction.

31. Determine the volume of the solid formed by revolving $y = 2 - x^2$ from $x = 0$ to $x = 1$ around the x-axis.

    $\frac{14\pi}{15}$. $\pi \int_0^1 (2-x^2)^2 dx$ evaluates to this value.

32. Determine the volume of the solid formed by revolving $x = 2y - y^2$ from $y = 0$ to $y = 1$ around the y-axis.

    $\frac{9\pi}{5}$. $\pi \int_0^1 (2y-y^2)^2 dy$ evaluates to this value.

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

    $V = \pi \int_c^d [f(y)]^2 \, dy$. Standard disc method formula for revolving around the y-axis.

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