Washer Method: Revolving Around Other Axes - AP Calculus BC
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Calculate the volume for $y = x^2$ and $y = x$ revolved around $y = 1$ from $x=0$ to $x=1$.
Calculate the volume for $y = x^2$ and $y = x$ revolved around $y = 1$ from $x=0$ to $x=1$.
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$V = \pi \int_{0}^{1} [(1-x^2)^2 - (1-x)^2] \ dx$. Both curves shifted by subtracting from 1 for revolution around $y=1$.
$V = \pi \int_{0}^{1} [(1-x^2)^2 - (1-x)^2] \ dx$. Both curves shifted by subtracting from 1 for revolution around $y=1$.
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Determine the outer radius for $x = 2$ revolved around $x = -3$.
Determine the outer radius for $x = 2$ revolved around $x = -3$.
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$R(y) = |2 - (-3)| = 5$. Distance from vertical line $x=2$ to axis $x=-3$ is $|2-(-3)|=5$.
$R(y) = |2 - (-3)| = 5$. Distance from vertical line $x=2$ to axis $x=-3$ is $|2-(-3)|=5$.
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Identify the axis of revolution for $y = f(x)$ revolved around $y = 2$.
Identify the axis of revolution for $y = f(x)$ revolved around $y = 2$.
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$y = 2$. The axis of revolution is the horizontal line around which the region rotates.
$y = 2$. The axis of revolution is the horizontal line around which the region rotates.
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What does $R(x)$ represent in the washer method formula?
What does $R(x)$ represent in the washer method formula?
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The outer radius of the washer. $R(x)$ is the distance from the axis of revolution to the farther curve.
The outer radius of the washer. $R(x)$ is the distance from the axis of revolution to the farther curve.
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What is the modified formula for revolving around a horizontal line $y = k$?
What is the modified formula for revolving around a horizontal line $y = k$?
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$V = \pi \int_{a}^{b} [(R(x)-k)^2 - (r(x)-k)^2] \ dx$. Shifts both radii by subtracting the axis position $k$ from each radius function.
$V = \pi \int_{a}^{b} [(R(x)-k)^2 - (r(x)-k)^2] \ dx$. Shifts both radii by subtracting the axis position $k$ from each radius function.
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What is the modified formula for revolving around a vertical line $x = k$?
What is the modified formula for revolving around a vertical line $x = k$?
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$V = \pi \int_{c}^{d} [(R(y)-k)^2 - (r(y)-k)^2] \ dy$. Integrates with respect to $y$ and shifts radii by the vertical axis position $k$.
$V = \pi \int_{c}^{d} [(R(y)-k)^2 - (r(y)-k)^2] \ dy$. Integrates with respect to $y$ and shifts radii by the vertical axis position $k$.
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How do you determine the outer radius $R(x)$ for $y = f(x)$ revolved around $y = k$?
How do you determine the outer radius $R(x)$ for $y = f(x)$ revolved around $y = k$?
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$R(x) = |f(x) - k|$. Distance from curve $f(x)$ to horizontal axis $y = k$ using absolute value.
$R(x) = |f(x) - k|$. Distance from curve $f(x)$ to horizontal axis $y = k$ using absolute value.
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How do you determine the inner radius $r(x)$ for $y = g(x)$ revolved around $y = k$?
How do you determine the inner radius $r(x)$ for $y = g(x)$ revolved around $y = k$?
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$r(x) = |g(x) - k|$. Distance from curve $g(x)$ to horizontal axis $y = k$ using absolute value.
$r(x) = |g(x) - k|$. Distance from curve $g(x)$ to horizontal axis $y = k$ using absolute value.
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Determine the outer radius for $x = 3$ revolving around $x = -1$.
Determine the outer radius for $x = 3$ revolving around $x = -1$.
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$R(y) = |3 - (-1)| = 4$. Distance from vertical line $x=3$ to axis $x=-1$ is $|3-(-1)|=4$.
$R(y) = |3 - (-1)| = 4$. Distance from vertical line $x=3$ to axis $x=-1$ is $|3-(-1)|=4$.
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State the bounds $a$ and $b$ for $y = f(x)$ from $x = 1$ to $x = 3$.
State the bounds $a$ and $b$ for $y = f(x)$ from $x = 1$ to $x = 3$.
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$a = 1$, $b = 3$. Integration bounds match the given $x$-interval from 1 to 3.
$a = 1$, $b = 3$. Integration bounds match the given $x$-interval from 1 to 3.
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For $x = 4$ revolving around $x = 0$, what is the outer radius?
For $x = 4$ revolving around $x = 0$, what is the outer radius?
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$R(y) = |4 - 0| = 4$. Distance from vertical line $x=4$ to y-axis is simply the absolute value 4.
$R(y) = |4 - 0| = 4$. Distance from vertical line $x=4$ to y-axis is simply the absolute value 4.
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When is the washer method preferred over the disk method?
When is the washer method preferred over the disk method?
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When there is an inner radius creating a hole. Washer method applies when the solid has a hollow center or gap.
When there is an inner radius creating a hole. Washer method applies when the solid has a hollow center or gap.
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Determine the outer radius for $x = 2$ revolved around $x = -3$.
Determine the outer radius for $x = 2$ revolved around $x = -3$.
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$R(y) = |2 - (-3)| = 5$. Distance from vertical line $x=2$ to axis $x=-3$ is $|2-(-3)|=5$.
$R(y) = |2 - (-3)| = 5$. Distance from vertical line $x=2$ to axis $x=-3$ is $|2-(-3)|=5$.
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In the washer method, how do you find the volume if $R(x) = x+1$ and $r(x) = x$?
In the washer method, how do you find the volume if $R(x) = x+1$ and $r(x) = x$?
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$V = \pi \int_{a}^{b} [(x+1)^2 - x^2] \ dx$. Standard washer formula with given outer and inner radius functions.
$V = \pi \int_{a}^{b} [(x+1)^2 - x^2] \ dx$. Standard washer formula with given outer and inner radius functions.
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What modification is needed for a vertical axis of revolution?
What modification is needed for a vertical axis of revolution?
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Integrate with respect to $y$ instead of $x$. Vertical axes require integrating with respect to $y$ instead of $x$.
Integrate with respect to $y$ instead of $x$. Vertical axes require integrating with respect to $y$ instead of $x$.
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Find the volume for $y = \sqrt{x}$ and $y = x$ revolved around $y = 3$ from $x=0$ to $x=1$.
Find the volume for $y = \sqrt{x}$ and $y = x$ revolved around $y = 3$ from $x=0$ to $x=1$.
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$V = \pi \int_{0}^{1} [(3-\sqrt{x})^2 - (3-x)^2] \ dx$. Both curves shifted by subtracting from 3 for revolution around $y=3$.
$V = \pi \int_{0}^{1} [(3-\sqrt{x})^2 - (3-x)^2] \ dx$. Both curves shifted by subtracting from 3 for revolution around $y=3$.
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Identify the outer radius for $x = 5$ revolved around $x = 0$.
Identify the outer radius for $x = 5$ revolved around $x = 0$.
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$R(y) = |5 - 0| = 5$. Distance from vertical line $x=5$ to y-axis ($x=0$) is simply 5.
$R(y) = |5 - 0| = 5$. Distance from vertical line $x=5$ to y-axis ($x=0$) is simply 5.
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Determine the outer radius for $y = f(x)$ revolved around $x = k$.
Determine the outer radius for $y = f(x)$ revolved around $x = k$.
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Calculate $R(y) = |f(y) - k|$. Distance from curve to vertical axis using absolute value for outer radius.
Calculate $R(y) = |f(y) - k|$. Distance from curve to vertical axis using absolute value for outer radius.
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What happens to $R(x)$ if the axis of revolution is above the graph?
What happens to $R(x)$ if the axis of revolution is above the graph?
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Increase $R(x)$ by the vertical distance. When axis is above the curve, add the vertical distance to get the radius.
Increase $R(x)$ by the vertical distance. When axis is above the curve, add the vertical distance to get the radius.
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Identify the inner radius for $y = 2x$ revolved around $y = -2$.
Identify the inner radius for $y = 2x$ revolved around $y = -2$.
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$r(x) = |2x - (-2)|$. Distance from line $y=2x$ to axis $y=-2$ using absolute value formula.
$r(x) = |2x - (-2)|$. Distance from line $y=2x$ to axis $y=-2$ using absolute value formula.
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If $R(x) = 5$ and $r(x) = 2$, what is the area of one washer?
If $R(x) = 5$ and $r(x) = 2$, what is the area of one washer?
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$A = \pi (5^2 - 2^2)$. Area of washer equals $\pi$ times outer radius squared minus inner radius squared.
$A = \pi (5^2 - 2^2)$. Area of washer equals $\pi$ times outer radius squared minus inner radius squared.
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What effect does a horizontal shift have on the washer method formula?
What effect does a horizontal shift have on the washer method formula?
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Adjust radii by the shift value in the formula. Horizontal shifts affect the distance calculations in the radius formulas.
Adjust radii by the shift value in the formula. Horizontal shifts affect the distance calculations in the radius formulas.
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Find the inner radius for $y = f(x)$ revolving around $x = c$ and $y = g(x)$.
Find the inner radius for $y = f(x)$ revolving around $x = c$ and $y = g(x)$.
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$r(y) = |g(y) - c|$. Distance from inner curve $g(y)$ to vertical axis $x=c$ using absolute value.
$r(y) = |g(y) - c|$. Distance from inner curve $g(y)$ to vertical axis $x=c$ using absolute value.
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What adjustment is made in the washer method when revolving around $x = k$?
What adjustment is made in the washer method when revolving around $x = k$?
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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.
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.
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State the corrected formula for revolving around a vertical line $x = k$.
State the corrected formula for revolving around a vertical line $x = k$.
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$V = \pi \int_{c}^{d} [(R(y)-k)^2 - (r(y)-k)^2] \ dy$. General formula for revolution around any vertical line at $x=k$.
$V = \pi \int_{c}^{d} [(R(y)-k)^2 - (r(y)-k)^2] \ dy$. General formula for revolution around any vertical line at $x=k$.
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State the formula for the volume of a solid using the washer method.
State the formula for the volume of a solid 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 squared minus inner radius squared.
$V = \pi \int_{a}^{b} [R(x)^2 - r(x)^2] \ dx$. Standard washer method formula with outer radius squared minus inner radius squared.
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How do you modify $R(x)$ and $r(x)$ for a shift along the y-axis?
How do you modify $R(x)$ and $r(x)$ for a shift along the y-axis?
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Adjust by the shift value in the $y$ direction. Add or subtract the shift distance from each radius function accordingly.
Adjust by the shift value in the $y$ direction. Add or subtract the shift distance from each radius function accordingly.
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Find the volume of $y = x^2$ and $y = 4$ revolved around $y = 5$ from $x=0$ to $x=2$.
Find the volume of $y = x^2$ and $y = 4$ revolved around $y = 5$ from $x=0$ to $x=2$.
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$V = \pi \int_{0}^{2} [(5-x^2)^2 - (5-4)^2] \ dx$. Outer radius is $5-x^2$, inner radius is $5-4=1$ for revolution around $y=5$.
$V = \pi \int_{0}^{2} [(5-x^2)^2 - (5-4)^2] \ dx$. Outer radius is $5-x^2$, inner radius is $5-4=1$ for revolution around $y=5$.
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Identify the formula for the solid's volume when $y = 3x$ and $y = 0$ are revolved around $y = -1$.
Identify the formula for the solid's volume when $y = 3x$ and $y = 0$ are revolved around $y = -1$.
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$V = \pi \int_{0}^{1} [(3x+1)^2 - 1^2] dx$. Both curves shifted by adding 1 for revolution around $y=-1$.
$V = \pi \int_{0}^{1} [(3x+1)^2 - 1^2] dx$. Both curves shifted by adding 1 for revolution around $y=-1$.
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For $R(x) = x^2$ and $r(x) = x$, what is the volume formula?
For $R(x) = x^2$ and $r(x) = x$, what is the volume formula?
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$V = \pi \int_{a}^{b} [x^4 - x^2] dx$. Outer radius squared is $x^4$, inner radius squared is $x^2$.
$V = \pi \int_{a}^{b} [x^4 - x^2] dx$. Outer radius squared is $x^4$, inner radius squared is $x^2$.
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