Washer Method: Revolving Around Other Axes - AP Calculus BC

1. State the bounds $a$ and $b$ for $y = f(x)$ from $x = 1$ to $x = 3$.

   $a = 1$, $b = 3$. Integration bounds match the given $x$-interval from 1 to 3.

2. What is the modified formula for revolving around a horizontal line $y = k$?

   $V = \pi \int_{a}^{b} [(R(x)-k)^2 - (r(x)-k)^2] \ dx$. Shifts both radii by subtracting the axis position $k$ from each radius function.

3. Identify the formula for the solid's volume when $y = 3x$ and $y = 0$ are revolved around $y = -1$.

   $V = \pi \int_{0}^{1} [(3x+1)^2 - 1^2] dx$. Both curves shifted by adding 1 for revolution around $y=-1$.

4. Find the inner radius for $y = f(x)$ revolving around $x = c$ and $y = g(x)$.

   $r(y) = |g(y) - c|$. Distance from inner curve $g(y)$ to vertical axis $x=c$ using absolute value.

5. Determine the outer radius for $x = 3$ revolving around $x = -1$.

   $R(y) = |3 - (-1)| = 4$. Distance from vertical line $x=3$ to axis $x=-1$ is $|3-(-1)|=4$.

6. Find the volume for $y = \sqrt{x}$ and $y = x$ revolved around $y = 3$ from $x=0$ to $x=1$.

   $V = \pi \int_{0}^{1} [(3-\sqrt{x})^2 - (3-x)^2] \ dx$. Both curves shifted by subtracting from 3 for revolution around $y=3$.

7. Determine the volume of $y = 3$ and $y = x^2$ revolved around $y = -1$ from $x=1$ to $x=2$.

   $V = \pi \int_{1}^{2} [(3+1)^2 - (x^2+1)^2] \ dx$. Shifts both curves by adding 1 to account for revolution around $y=-1$.

8. How do you determine the outer radius $R(x)$ for $y = f(x)$ revolved around $y = k$?

   $R(x) = |f(x) - k|$. Distance from curve $f(x)$ to horizontal axis $y = k$ using absolute value.

9. Identify the inner radius for $y = 2x$ revolved around $y = -2$.

   $r(x) = |2x - (-2)|$. Distance from line $y=2x$ to axis $y=-2$ using absolute value formula.

10. What happens to $R(x)$ if the axis of revolution is above the graph?

    Increase $R(x)$ by the vertical distance. When axis is above the curve, add the vertical distance to get the radius.

11. Determine the outer radius for $y = f(x)$ revolved around $x = k$.

    Calculate $R(y) = |f(y) - k|$. Distance from curve to vertical axis using absolute value for outer radius.

12. What does $R(x)$ represent in the washer method formula?

    The outer radius of the washer. $R(x)$ is the distance from the axis of revolution to the farther curve.

13. Calculate the volume for $y = x^2$ and $y = x$ revolved around $y = 1$ from $x=0$ to $x=1$.

    $V = \pi \int_{0}^{1} [(1-x^2)^2 - (1-x)^2] \ dx$. Both curves shifted by subtracting from 1 for revolution around $y=1$.

14. When is the washer method preferred over the disk method?

    When there is an inner radius creating a hole. Washer method applies when the solid has a hollow center or gap.

15. For $x = 4$ revolving around $x = 0$, what is the outer radius?

    $R(y) = |4 - 0| = 4$. Distance from vertical line $x=4$ to y-axis is simply the absolute value 4.

16. What is the integral setup for $y = x$ and $y = 0$ revolved around $y = 2$?

    $V = \pi \int_{0}^{1} [(2-x)^2 - (2)^2] \ dx$. Outer radius is $2-x$, inner radius is $2-0=2$ for revolution around $y=2$.

17. State the corrected formula for revolving around a vertical line $x = k$.

    $V = \pi \int_{c}^{d} [(R(y)-k)^2 - (r(y)-k)^2] \ dy$. General formula for revolution around any vertical line at $x=k$.

18. In the washer method, how do you find the volume if $R(x) = x+1$ and $r(x) = x$?

    $V = \pi \int_{a}^{b} [(x+1)^2 - x^2] \ dx$. Standard washer formula with given outer and inner radius functions.

19. How do you determine the inner radius $r(x)$ for $y = g(x)$ revolved around $y = k$?

    $r(x) = |g(x) - k|$. Distance from curve $g(x)$ to horizontal axis $y = k$ using absolute value.

20. State the formula for the volume of a solid using the washer method.

    $V = \pi \int_{a}^{b} [R(x)^2 - r(x)^2] \ dx$. Standard washer method formula with outer radius squared minus inner radius squared.

21. For $R(x) = x^2$ and $r(x) = x$, what is the volume formula?

    $V = \pi \int_{a}^{b} [x^4 - x^2] dx$. Outer radius squared is $x^4$, inner radius squared is $x^2$.

22. Identify the axis of revolution for $y = f(x)$ revolved around $y = 2$.

    $y = 2$. The axis of revolution is the horizontal line around which the region rotates.

23. Find the volume of $y = x^2$ and $y = 4$ revolved around $y = 5$ from $x=0$ to $x=2$.

    $V = \pi \int_{0}^{2} [(5-x^2)^2 - (5-4)^2] \ dx$. Outer radius is $5-x^2$, inner radius is $5-4=1$ for revolution around $y=5$.

24. If $R(x) = 5$ and $r(x) = 2$, what is the area of one washer?

    $A = \pi (5^2 - 2^2)$. Area of washer equals $\pi$ times outer radius squared minus inner radius squared.

25. What modification is needed for a vertical axis of revolution?

    Integrate with respect to $y$ instead of $x$. Vertical axes require integrating with respect to $y$ instead of $x$.

26. What is the primary advantage of the washer method?

    Accommodates hollow regions in solids of revolution. Handles solids with holes or gaps, unlike the basic disk method.

27. Identify the outer radius for $x = 5$ revolved around $x = 0$.

    $R(y) = |5 - 0| = 5$. Distance from vertical line $x=5$ to y-axis ($x=0$) is simply 5.

28. How do you modify $R(x)$ and $r(x)$ for a shift along the y-axis?

    Adjust by the shift value in the $y$ direction. Add or subtract the shift distance from each radius function accordingly.

29. What adjustment is made in the washer method when revolving around $x = k$?

    Use $R(y)$ and $r(y)$, integrate with respect to $y$. Switch to functions of $y$ and integrate with respect to $dy$ for vertical axes.

30. What is the modified formula for revolving around a vertical line $x = k$?

    $V = \pi \int_{c}^{d} [(R(y)-k)^2 - (r(y)-k)^2] \ dy$. Integrates with respect to $y$ and shifts radii by the vertical axis position $k$.

31. What is the integral setup for $y = \sqrt{x}$ and $y = 2$ revolved around $y = 0$?

    $V = \pi \int_{0}^{4} [(2)^2 - (\sqrt{x})^2] \ dx$. Outer radius is 2, inner radius is $\sqrt{x}$ for revolution around x-axis.

32. What is the purpose of the washer method in calculus?

    To find the volume of solids of revolution with gaps. Creates volumes with hollow centers by subtracting inner from outer areas.

33. Determine the outer radius for $x = 2$ revolved around $x = -3$.

    $R(y) = |2 - (-3)| = 5$. Distance from vertical line $x=2$ to axis $x=-3$ is $|2-(-3)|=5$.

34. What effect does a horizontal shift have on the washer method formula?

    Adjust radii by the shift value in the formula. Horizontal shifts affect the distance calculations in the radius formulas.

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