Washer Method: Revolving Around Other Axes - AP Calculus AB
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How to set $R(x)$ if revolving around $y = 0$, $y = x^2$?
How to set $R(x)$ if revolving around $y = 0$, $y = x^2$?
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$R(x) = x^2$. Distance from $y = x^2$ to the $x$-axis ($y = 0$) is $x^2$.
$R(x) = x^2$. Distance from $y = x^2$ to the $x$-axis ($y = 0$) is $x^2$.
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How do you find the inner radius for revolution around $y$-axis?
How do you find the inner radius for revolution around $y$-axis?
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Distance from the $y$-axis to the nearest curve. Take the minimum $x$-coordinate of the curves being revolved.
Distance from the $y$-axis to the nearest curve. Take the minimum $x$-coordinate of the curves being revolved.
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Identify the axis of revolution in $y = f(x)$ revolved around $y = c$.
Identify the axis of revolution in $y = f(x)$ revolved around $y = c$.
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$y = c$. The horizontal line about which the region is rotated.
$y = c$. The horizontal line about which the region is rotated.
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Identify $r(x)$ if $y = x$ revolves around $y = -1$.
Identify $r(x)$ if $y = x$ revolves around $y = -1$.
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$r(x) = x + 1$. Distance from $y = x$ to horizontal line $y = -1$ is $x - (-1) = x + 1$.
$r(x) = x + 1$. Distance from $y = x$ to horizontal line $y = -1$ is $x - (-1) = x + 1$.
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Which axis is used in $x = g(y)$ revolved around $x = c$?
Which axis is used in $x = g(y)$ revolved around $x = c$?
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$x = c$. The vertical line about which the region is rotated.
$x = c$. The vertical line about which the region is rotated.
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Find $r(x)$ if revolving around $y = 2$, $g(x) = x$.
Find $r(x)$ if revolving around $y = 2$, $g(x) = x$.
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$r(x) = 2 - x$. Distance from $y = x$ to the line $y = 2$ is $2 - x$.
$r(x) = 2 - x$. Distance from $y = x$ to the line $y = 2$ is $2 - x$.
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State the limits of integration if revolving around $y=c$ for $x = f(y)$.
State the limits of integration if revolving around $y=c$ for $x = f(y)$.
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From $y = a$ to $y = b$. Integration bounds match the $y$-values where the region is defined.
From $y = a$ to $y = b$. Integration bounds match the $y$-values where the region is defined.
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Determine $R(y)$ for revolution around $x = -1$, $x = y^2 + 1$.
Determine $R(y)$ for revolution around $x = -1$, $x = y^2 + 1$.
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$R(y) = y^2 + 2$. Distance from $x = y^2 + 1$ to line $x = -1$ is $(y^2 + 1) - (-1) = y^2 + 2$.
$R(y) = y^2 + 2$. Distance from $x = y^2 + 1$ to line $x = -1$ is $(y^2 + 1) - (-1) = y^2 + 2$.
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Find $R(x)$ if revolving around $y = 2$, $f(x) = x^2$.
Find $R(x)$ if revolving around $y = 2$, $f(x) = x^2$.
Tap to reveal answer
$R(x) = 2 - x^2$. Distance from $y = x^2$ to the line $y = 2$ is $2 - x^2$.
$R(x) = 2 - x^2$. Distance from $y = x^2$ to the line $y = 2$ is $2 - x^2$.
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Identify $R(x)$ if $y = x^2$ revolves around $y = -1$.
Identify $R(x)$ if $y = x^2$ revolves around $y = -1$.
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$R(x) = x^2 + 1$. Distance from $y = x^2$ to horizontal line $y = -1$ is $x^2 - (-1) = x^2 + 1$.
$R(x) = x^2 + 1$. Distance from $y = x^2$ to horizontal line $y = -1$ is $x^2 - (-1) = x^2 + 1$.
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What is the area of the washer if $R = 4$ and $r = 2$?
What is the area of the washer if $R = 4$ and $r = 2$?
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$\pi (16 - 4)$. Area of washer is $\pi(R^2 - r^2) = \pi(16 - 4) = 12\pi$.
$\pi (16 - 4)$. Area of washer is $\pi(R^2 - r^2) = \pi(16 - 4) = 12\pi$.
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Determine $r(y)$ for revolution around $x = -1$, $x = y + 1$.
Determine $r(y)$ for revolution around $x = -1$, $x = y + 1$.
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$r(y) = y + 2$. Distance from $x = y + 1$ to line $x = -1$ is $(y + 1) - (-1) = y + 2$.
$r(y) = y + 2$. Distance from $x = y + 1$ to line $x = -1$ is $(y + 1) - (-1) = y + 2$.
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What does $R(x)$ represent in the washer method?
What does $R(x)$ represent in the washer method?
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The outer radius function. The distance from the axis of revolution to the farther boundary curve.
The outer radius function. The distance from the axis of revolution to the farther boundary curve.
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State the formula for volume using the washer method.
State the formula for volume using the washer method.
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$V = \pi \int_a^b , [(R(x))^2 - (r(x))^2] , dx$. Standard washer method formula with outer radius $R(x)$ and inner radius $r(x)$.
$V = \pi \int_a^b , [(R(x))^2 - (r(x))^2] , dx$. Standard washer method formula with outer radius $R(x)$ and inner radius $r(x)$.
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What does $R(x)$ represent in the washer method?
What does $R(x)$ represent in the washer method?
Tap to reveal answer
The outer radius function. The distance from the axis of revolution to the farther boundary curve.
The outer radius function. The distance from the axis of revolution to the farther boundary curve.
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State the formula for volume using the washer method.
State the formula for volume using the washer method.
Tap to reveal 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 $r(x)$.
$V = \pi \int_a^b , [(R(x))^2 - (r(x))^2] , dx$. Standard washer method formula with outer radius $R(x)$ and inner radius $r(x)$.
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How to set $R(x)$ if revolving around $y = 0$, $y = x^2$?
How to set $R(x)$ if revolving around $y = 0$, $y = x^2$?
Tap to reveal answer
$R(x) = x^2$. Distance from $y = x^2$ to the $x$-axis ($y = 0$) is $x^2$.
$R(x) = x^2$. Distance from $y = x^2$ to the $x$-axis ($y = 0$) is $x^2$.
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Identify $r(x)$ if $y = x$ revolves around $y = -1$.
Identify $r(x)$ if $y = x$ revolves around $y = -1$.
Tap to reveal answer
$r(x) = x + 1$. Distance from $y = x$ to horizontal line $y = -1$ is $x - (-1) = x + 1$.
$r(x) = x + 1$. Distance from $y = x$ to horizontal line $y = -1$ is $x - (-1) = x + 1$.
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Determine $r(y)$ for revolution around $x = -1$, $x = y + 1$.
Determine $r(y)$ for revolution around $x = -1$, $x = y + 1$.
Tap to reveal answer
$r(y) = y + 2$. Distance from $x = y + 1$ to line $x = -1$ is $(y + 1) - (-1) = y + 2$.
$r(y) = y + 2$. Distance from $x = y + 1$ to line $x = -1$ is $(y + 1) - (-1) = y + 2$.
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What is the area of the washer if $R = 4$ and $r = 2$?
What is the area of the washer if $R = 4$ and $r = 2$?
Tap to reveal answer
$\pi (16 - 4)$. Area of washer is $\pi(R^2 - r^2) = \pi(16 - 4) = 12\pi$.
$\pi (16 - 4)$. Area of washer is $\pi(R^2 - r^2) = \pi(16 - 4) = 12\pi$.
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Identify $R(x)$ if $y = x^2$ revolves around $y = -1$.
Identify $R(x)$ if $y = x^2$ revolves around $y = -1$.
Tap to reveal answer
$R(x) = x^2 + 1$. Distance from $y = x^2$ to horizontal line $y = -1$ is $x^2 - (-1) = x^2 + 1$.
$R(x) = x^2 + 1$. Distance from $y = x^2$ to horizontal line $y = -1$ is $x^2 - (-1) = x^2 + 1$.
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Which axis is used in $x = g(y)$ revolved around $x = c$?
Which axis is used in $x = g(y)$ revolved around $x = c$?
Tap to reveal answer
$x = c$. The vertical line about which the region is rotated.
$x = c$. The vertical line about which the region is rotated.
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What does $r(x)$ represent in the washer method?
What does $r(x)$ represent in the washer method?
Tap to reveal answer
The inner radius function. The distance from the axis of revolution to the nearer boundary curve.
The inner radius function. The distance from the axis of revolution to the nearer boundary curve.
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Identify the axis of revolution in $y = f(x)$ revolved around $y = c$.
Identify the axis of revolution in $y = f(x)$ revolved around $y = c$.
Tap to reveal answer
$y = c$. The horizontal line about which the region is rotated.
$y = c$. The horizontal line about which the region is rotated.
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How do you find the outer radius for revolution around $y$-axis?
How do you find the outer radius for revolution around $y$-axis?
Tap to reveal answer
Distance from the $y$-axis to the farthest curve. Take the maximum $x$-coordinate of the curves being revolved.
Distance from the $y$-axis to the farthest curve. Take the maximum $x$-coordinate of the curves being revolved.
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How do you find the inner radius for revolution around $y$-axis?
How do you find the inner radius for revolution around $y$-axis?
Tap to reveal answer
Distance from the $y$-axis to the nearest curve. Take the minimum $x$-coordinate of the curves being revolved.
Distance from the $y$-axis to the nearest curve. Take the minimum $x$-coordinate of the curves being revolved.
← Didn't Know|Knew It →
Find $R(x)$ if revolving around $y = 2$, $f(x) = x^2$.
Find $R(x)$ if revolving around $y = 2$, $f(x) = x^2$.
Tap to reveal answer
$R(x) = 2 - x^2$. Distance from $y = x^2$ to the line $y = 2$ is $2 - x^2$.
$R(x) = 2 - x^2$. Distance from $y = x^2$ to the line $y = 2$ is $2 - x^2$.
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Determine $R(y)$ for revolution around $x = -1$, $x = y^2 + 1$.
Determine $R(y)$ for revolution around $x = -1$, $x = y^2 + 1$.
Tap to reveal answer
$R(y) = y^2 + 2$. Distance from $x = y^2 + 1$ to line $x = -1$ is $(y^2 + 1) - (-1) = y^2 + 2$.
$R(y) = y^2 + 2$. Distance from $x = y^2 + 1$ to line $x = -1$ is $(y^2 + 1) - (-1) = y^2 + 2$.
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State the limits of integration if revolving around $y=c$ for $x = f(y)$.
State the limits of integration if revolving around $y=c$ for $x = f(y)$.
Tap to reveal answer
From $y = a$ to $y = b$. Integration bounds match the $y$-values where the region is defined.
From $y = a$ to $y = b$. Integration bounds match the $y$-values where the region is defined.
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Find $r(x)$ if revolving around $y = 2$, $g(x) = x$.
Find $r(x)$ if revolving around $y = 2$, $g(x) = x$.
Tap to reveal answer
$r(x) = 2 - x$. Distance from $y = x$ to the line $y = 2$ is $2 - x$.
$r(x) = 2 - x$. Distance from $y = x$ to the line $y = 2$ is $2 - x$.
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