AP Calculus AB Flashcards: Washer Method Revolving Around Other Axes

What this deck covers

This deck focuses on Washer Method Revolving Around Other Axes, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

Flashcard 3: Determine $r(y)$ for revolution around $x = -1$, $x = y + 1$.

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

Flashcard 4: Find $r(x)$ if revolving around $y = 2$, $g(x) = x$.

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

Flashcard 5: How to set $R(x)$ if revolving around $y = 0$, $y = x^2$?

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

Flashcard 6: How do you find the outer radius for revolution around $y$-axis?

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

Flashcard 7: Identify $R(x)$ if $y = x^2$ revolves around $y = -1$.

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

Flashcard 8: Identify $r(x)$ if $y = x$ revolves around $y = -1$.

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

Flashcard 9: How do you find the inner radius for revolution around $y$-axis?

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

Flashcard 10: What is the area of the washer if $R = 4$ and $r = 2$?

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

Flashcard 11: Identify the axis of revolution in $y = f(x)$ revolved around $y = c$.

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

Flashcard 12: How do you find the outer radius for revolution around $y$-axis?

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

Flashcard 13: Which axis is used in $x = g(y)$ revolved around $x = c$?

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

Flashcard 14: State the formula for volume using the washer method.

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

Flashcard 15: Determine $R(y)$ for revolution around $x = -1$, $x = y^2 + 1$.

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

Flashcard 16: Find $R(x)$ if revolving around $y = 2$, $f(x) = x^2$.

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

Flashcard 17: Determine $r(y)$ for revolution around $x = -1$, $x = y + 1$.

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

Flashcard 18: State the formula for volume using the washer method.

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

Flashcard 19: Identify the axis of revolution in $y = f(x)$ revolved around $y = c$.

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

Flashcard 20: Identify $r(x)$ if $y = x$ revolves around $y = -1$.

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

Flashcard 21: State the limits of integration if revolving around $y=c$ for $x = f(y)$.

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

Flashcard 22: How do you find the inner radius for revolution around $y$-axis?

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

Flashcard 23: Which axis is used in $x = g(y)$ revolved around $x = c$?

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

Flashcard 24: State the limits of integration if revolving around $y=c$ for $x = f(y)$.

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

Flashcard 25: What does $r(x)$ represent in the washer method?

Answer: The inner radius function. The distance from the axis of revolution to the nearer boundary curve.

Flashcard 26: How to set $R(x)$ if revolving around $y = 0$, $y = x^2$?

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

Flashcard 27: What is the area of the washer if $R = 4$ and $r = 2$?

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

Flashcard 28: What does $R(x)$ represent in the washer method?

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

Flashcard 29: What does $R(x)$ represent in the washer method?

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

Flashcard 30: Find $R(x)$ if revolving around $y = 2$, $f(x) = x^2$.

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

Flashcard 31: Find $r(x)$ if revolving around $y = 2$, $g(x) = x$.

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