Disc Method: Revolving Around x/y Axes - AP Calculus BC
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Calculate the volume of the solid formed by revolving $y = \frac{1}{2}x$ from $x = 0$ to $x = 3$ around the x-axis.
Calculate the volume of the solid formed by revolving $y = \frac{1}{2}x$ from $x = 0$ to $x = 3$ around the x-axis.
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$\frac{9\pi}{8}$. $\pi \int_0^3 (\frac{x}{2})^2 dx = \pi \int_0^3 \frac{x^2}{4} dx$.
$\frac{9\pi}{8}$. $\pi \int_0^3 (\frac{x}{2})^2 dx = \pi \int_0^3 \frac{x^2}{4} dx$.
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Identify the axis of revolution for the formula: $V = \pi \int_a^b [f(y)]^2 , dy$.
Identify the axis of revolution for the formula: $V = \pi \int_a^b [f(y)]^2 , dy$.
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y-axis. Variable $y$ in the integral indicates y-axis revolution.
y-axis. Variable $y$ in the integral indicates y-axis revolution.
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Identify the volume formula for the solid formed by revolving $y = x$ from $x = 0$ to $x = 2$ around x-axis.
Identify the volume formula for the solid formed by revolving $y = x$ from $x = 0$ to $x = 2$ around x-axis.
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$V = \pi \int_0^2 x^2 , dx$. Radius is $f(x) = x$, so we integrate $\pi x^2$.
$V = \pi \int_0^2 x^2 , dx$. Radius is $f(x) = x$, so we integrate $\pi x^2$.
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Determine the volume generated by revolving $y = 1 - x^2$ from $x = -1$ to $x = 1$ around the x-axis.
Determine the volume generated by revolving $y = 1 - x^2$ from $x = -1$ to $x = 1$ around the x-axis.
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$\frac{16\pi}{15}$. $\pi \int_{-1}^1 (1-x^2)^2 dx$ evaluates to this value.
$\frac{16\pi}{15}$. $\pi \int_{-1}^1 (1-x^2)^2 dx$ evaluates to this value.
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Identify the function to be squared in the integral for the disc method around the x-axis.
Identify the function to be squared in the integral for the disc method around the x-axis.
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$f(x)$. For x-axis revolution, the radius function is $f(x)$.
$f(x)$. For x-axis revolution, the radius function is $f(x)$.
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What does the variable $d$ represent in the formula $V = \pi \int_c^d [f(y)]^2 , dy$?
What does the variable $d$ represent in the formula $V = \pi \int_c^d [f(y)]^2 , dy$?
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Upper bound of integration along y-axis. Ending point of integration interval for y-axis.
Upper bound of integration along y-axis. Ending point of integration interval for y-axis.
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What represents the height of the disc in the disc method?
What represents the height of the disc in the disc method?
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Infinitesimal thickness $dx$ or $dy$. Represents the width of each infinitesimal disc slice.
Infinitesimal thickness $dx$ or $dy$. Represents the width of each infinitesimal disc slice.
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Identify the volume formula for the solid formed by revolving $x = y^2$ from $y = 0$ to $y = 1$ around y-axis.
Identify the volume formula for the solid formed by revolving $x = y^2$ from $y = 0$ to $y = 1$ around y-axis.
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$V = \pi \int_0^1 y^4 , dy$. Radius is $f(y) = y^2$, so we integrate $\pi y^4$.
$V = \pi \int_0^1 y^4 , dy$. Radius is $f(y) = y^2$, so we integrate $\pi y^4$.
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Calculate the volume of the solid formed by revolving $y = \cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ around the x-axis.
Calculate the volume of the solid formed by revolving $y = \cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ around the x-axis.
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$\frac{\pi^2}{4}$. $\pi \int_0^{\pi/2} \cos^2(x) dx$ using power reduction.
$\frac{\pi^2}{4}$. $\pi \int_0^{\pi/2} \cos^2(x) dx$ using power reduction.
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What type of integral is used in the disc method?
What type of integral is used in the disc method?
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Definite integral. Volume calculations require definite integration.
Definite integral. Volume calculations require definite integration.
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What is the volume of the solid formed by revolving $x = 3$ from $y = 0$ to $y = 2$ around the y-axis?
What is the volume of the solid formed by revolving $x = 3$ from $y = 0$ to $y = 2$ around the y-axis?
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$18\pi$. $\pi \int_0^2 3^2 dy = \pi \int_0^2 9 dy = 18\pi$.
$18\pi$. $\pi \int_0^2 3^2 dy = \pi \int_0^2 9 dy = 18\pi$.
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Determine the volume of the solid formed by revolving $x = 2y - y^2$ from $y = 0$ to $y = 1$ around the y-axis.
Determine the volume of the solid formed by revolving $x = 2y - y^2$ from $y = 0$ to $y = 1$ around the y-axis.
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$\frac{9\pi}{5}$. $\pi \int_0^1 (2y-y^2)^2 dy$ evaluates to this value.
$\frac{9\pi}{5}$. $\pi \int_0^1 (2y-y^2)^2 dy$ evaluates to this value.
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State the formula for volume using the disc method around the y-axis.
State the formula for volume using the disc method around the y-axis.
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$V = \pi \int_c^d [f(y)]^2 , dy$. Standard disc method formula for revolving around the y-axis.
$V = \pi \int_c^d [f(y)]^2 , dy$. Standard disc method formula for revolving around the y-axis.
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State the formula for volume using the disc method around the x-axis.
State the formula for volume using the disc method around the x-axis.
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$V = \pi \int_a^b [f(x)]^2 , dx$. Standard disc method formula for revolving around the x-axis.
$V = \pi \int_a^b [f(x)]^2 , dx$. Standard disc method formula for revolving around the x-axis.
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Define the disc method in calculus.
Define the disc method in calculus.
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A method to find volume by revolving a region around an axis. Creates circular cross-sections perpendicular to the axis.
A method to find volume by revolving a region around an axis. Creates circular cross-sections perpendicular to the axis.
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What is the result of integrating $\pi [f(x)]^2$ with respect to $x$ from $a$ to $b$?
What is the result of integrating $\pi [f(x)]^2$ with respect to $x$ from $a$ to $b$?
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Volume of solid of revolution. Disc method integration gives volume of revolution.
Volume of solid of revolution. Disc method integration gives volume of revolution.
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Determine the volume of the solid formed by revolving $y = x^2$ from $x = 0$ to $x = 1$ around the x-axis.
Determine the volume of the solid formed by revolving $y = x^2$ from $x = 0$ to $x = 1$ around the x-axis.
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$\frac{\pi}{5}$. $\pi \int_0^1 x^4 dx = \pi[\frac{x^5}{5}]_0^1 = \frac{\pi}{5}$.
$\frac{\pi}{5}$. $\pi \int_0^1 x^4 dx = \pi[\frac{x^5}{5}]_0^1 = \frac{\pi}{5}$.
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What is the radius of the disc in the disc method when revolving $y = f(x)$ around the x-axis?
What is the radius of the disc in the disc method when revolving $y = f(x)$ around the x-axis?
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$f(x)$. Distance from x-axis to the curve is $f(x)$.
$f(x)$. Distance from x-axis to the curve is $f(x)$.
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When revolving around the y-axis, what represents the height of the disc?
When revolving around the y-axis, what represents the height of the disc?
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Infinitesimal thickness $dy$. Width of each disc slice when integrating along y-axis.
Infinitesimal thickness $dy$. Width of each disc slice when integrating along y-axis.
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What does the variable $b$ represent in the formula $V = \pi \int_a^b [f(x)]^2 , dx$?
What does the variable $b$ represent in the formula $V = \pi \int_a^b [f(x)]^2 , dx$?
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Upper bound of integration along x-axis. Ending point of integration interval for x-axis.
Upper bound of integration along x-axis. Ending point of integration interval for x-axis.
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Calculate the volume of the solid formed by revolving $y = \sin(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ around the x-axis.
Calculate the volume of the solid formed by revolving $y = \sin(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ around the x-axis.
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$\frac{\pi^2}{4}$. $\pi \int_0^{\pi/2} \sin^2(x) dx$ using power reduction.
$\frac{\pi^2}{4}$. $\pi \int_0^{\pi/2} \sin^2(x) dx$ using power reduction.
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Identify the function to be squared in the integral for the disc method around the y-axis.
Identify the function to be squared in the integral for the disc method around the y-axis.
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$f(y)$. For y-axis revolution, the radius function is $f(y)$.
$f(y)$. For y-axis revolution, the radius function is $f(y)$.
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What does the variable $c$ represent in the formula $V = \pi \int_c^d [f(y)]^2 , dy$?
What does the variable $c$ represent in the formula $V = \pi \int_c^d [f(y)]^2 , dy$?
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Lower bound of integration along y-axis. Starting point of integration interval for y-axis.
Lower bound of integration along y-axis. Starting point of integration interval for y-axis.
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What is the radius of the disc in the disc method when revolving $x = g(y)$ around the y-axis?
What is the radius of the disc in the disc method when revolving $x = g(y)$ around the y-axis?
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$g(y)$. Distance from y-axis to the curve is $g(y)$.
$g(y)$. Distance from y-axis to the curve is $g(y)$.
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What is the result of integrating $\pi [g(y)]^2$ with respect to $y$ from $c$ to $d$?
What is the result of integrating $\pi [g(y)]^2$ with respect to $y$ from $c$ to $d$?
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Volume of solid of revolution. Disc method integration gives volume of revolution.
Volume of solid of revolution. Disc method integration gives volume of revolution.
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What is the integral limit when revolving region from $x = 1$ to $x = 3$ around x-axis?
What is the integral limit when revolving region from $x = 1$ to $x = 3$ around x-axis?
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$1$ to $3$. Limits match the given x-interval bounds.
$1$ to $3$. Limits match the given x-interval bounds.
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State the formula for volume using the disc method around the x-axis.
State the formula for volume using the disc method around the x-axis.
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$V = \pi \int_a^b [f(x)]^2 , dx$. Standard disc method formula for revolving around the x-axis.
$V = \pi \int_a^b [f(x)]^2 , dx$. Standard disc method formula for revolving around the x-axis.
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What is the volume of the solid formed by revolving $x = 3$ from $y = 0$ to $y = 2$ around the y-axis?
What is the volume of the solid formed by revolving $x = 3$ from $y = 0$ to $y = 2$ around the y-axis?
Tap to reveal answer
$18\pi$. $\pi \int_0^2 3^2 dy = \pi \int_0^2 9 dy = 18\pi$.
$18\pi$. $\pi \int_0^2 3^2 dy = \pi \int_0^2 9 dy = 18\pi$.
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Identify the function to be squared in the integral for the disc method around the x-axis.
Identify the function to be squared in the integral for the disc method around the x-axis.
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$f(x)$. For x-axis revolution, the radius function is $f(x)$.
$f(x)$. For x-axis revolution, the radius function is $f(x)$.
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What type of integral is used in the disc method?
What type of integral is used in the disc method?
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Definite integral. Volume calculations require definite integration.
Definite integral. Volume calculations require definite integration.
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