Washer Method: Revolving Around Other Axes - AP Calculus AB

1. Identify $R(x)$ if $y = x^2$ revolves around $y = -1$.

   $R(x) = x^2 + 1$. Distance from $y = x^2$ to horizontal line $y = -1$ is $x^2 - (-1) = x^2 + 1$.

2. How to set $R(x)$ if revolving around $y = 0$, $y = x^2$?

   $R(x) = x^2$. Distance from $y = x^2$ to the $x$-axis ($y = 0$) is $x^2$.

3. How do you find the outer radius for revolution around $y$-axis?

   Distance from the $y$-axis to the farthest curve. Take the maximum $x$-coordinate of the curves being revolved.

4. Which axis is used in $x = g(y)$ revolved around $x = c$?

   $x = c$. The vertical line about which the region is rotated.

5. Determine $R(y)$ for revolution around $x = -1$, $x = y^2 + 1$.

   $R(y) = y^2 + 2$. Distance from $x = y^2 + 1$ to line $x = -1$ is $(y^2 + 1) - (-1) = y^2 + 2$.

6. Find $R(x)$ if revolving around $y = 2$, $f(x) = x^2$.

   $R(x) = 2 - x^2$. Distance from $y = x^2$ to the line $y = 2$ is $2 - x^2$.

7. Determine $r(y)$ for revolution around $x = -1$, $x = y + 1$.

   $r(y) = y + 2$. Distance from $x = y + 1$ to line $x = -1$ is $(y + 1) - (-1) = y + 2$.

8. State the formula for volume using the washer method.

   $V = \pi \int_a^b \, [(R(x))^2 - (r(x))^2] \, dx$. Standard washer method formula with outer radius $R(x)$ and inner radius $r(x)$.

9. Identify the axis of revolution in $y = f(x)$ revolved around $y = c$.

   $y = c$. The horizontal line about which the region is rotated.

10. Identify $r(x)$ if $y = x$ revolves around $y = -1$.

    $r(x) = x + 1$. Distance from $y = x$ to horizontal line $y = -1$ is $x - (-1) = x + 1$.

11. State the limits of integration if revolving around $y=c$ for $x = f(y)$.

    From $y = a$ to $y = b$. Integration bounds match the $y$-values where the region is defined.

12. How do you find the inner radius for revolution around $y$-axis?

    Distance from the $y$-axis to the nearest curve. Take the minimum $x$-coordinate of the curves being revolved.

13. What is the area of the washer if $R = 4$ and $r = 2$?

    $\pi (16 - 4)$. Area of washer is $\pi(R^2 - r^2) = \pi(16 - 4) = 12\pi$.

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

    The outer radius function. The distance from the axis of revolution to the farther boundary curve.

15. Find $r(x)$ if revolving around $y = 2$, $g(x) = x$.

    $r(x) = 2 - x$. Distance from $y = x$ to the line $y = 2$ is $2 - x$.

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