Disc Method: Revolving Around x/y Axes - AP Calculus AB
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What does $g(y)$ represent in the disc method formula $V = \int_c^d \pi [g(y)]^2 , dy$?
What does $g(y)$ represent in the disc method formula $V = \int_c^d \pi [g(y)]^2 , dy$?
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Radius of the disc. Distance from function to y-axis forms the disc radius.
Radius of the disc. Distance from function to y-axis forms the disc radius.
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Find the volume of the solid obtained by revolving $y=\frac{1}{x}$ from $x=1$ to $x=2$ around the x-axis.
Find the volume of the solid obtained by revolving $y=\frac{1}{x}$ from $x=1$ to $x=2$ around the x-axis.
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$V = \frac{\pi}{2}$. $V = \pi \int_1^2 \frac{1}{x^2} dx = \pi [-\frac{1}{x}]_1^2 = \frac{\pi}{2}$
$V = \frac{\pi}{2}$. $V = \pi \int_1^2 \frac{1}{x^2} dx = \pi [-\frac{1}{x}]_1^2 = \frac{\pi}{2}$
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What is the role of the function $g(y)$ in the disc method when revolving around the y-axis?
What is the role of the function $g(y)$ in the disc method when revolving around the y-axis?
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Defines the radius of discs. Function determines how far each disc extends from the axis.
Defines the radius of discs. Function determines how far each disc extends from the axis.
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What is the cross-sectional area of a disc with radius $r$?
What is the cross-sectional area of a disc with radius $r$?
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$\pi r^2$. Standard formula for area of a circle.
$\pi r^2$. Standard formula for area of a circle.
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Given $y=f(x)$, identify the limits of integration when revolving from $x=a$ to $x=b$ around the x-axis.
Given $y=f(x)$, identify the limits of integration when revolving from $x=a$ to $x=b$ around the x-axis.
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$a$ to $b$. Limits match the given x-interval for the region.
$a$ to $b$. Limits match the given x-interval for the region.
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What does $f(x)$ represent in the disc method formula $V = \int_a^b \pi [f(x)]^2 , dx$?
What does $f(x)$ represent in the disc method formula $V = \int_a^b \pi [f(x)]^2 , dx$?
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Radius of the disc. Distance from function to x-axis forms the disc radius.
Radius of the disc. Distance from function to x-axis forms the disc radius.
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What is the integral setup for a disc method problem revolving $y=x^3$ from $x=0$ to $x=1$?
What is the integral setup for a disc method problem revolving $y=x^3$ from $x=0$ to $x=1$?
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$V = \int_0^1 \pi (x^3)^2 , dx$. Disc method setup with radius $x^3$ squared in integrand.
$V = \int_0^1 \pi (x^3)^2 , dx$. Disc method setup with radius $x^3$ squared in integrand.
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Find the volume of the solid obtained by revolving $y=5$ from $x=0$ to $x=2$ around the x-axis.
Find the volume of the solid obtained by revolving $y=5$ from $x=0$ to $x=2$ around the x-axis.
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$V = 50\pi$. $V = \pi \int_0^2 25 dx = 25\pi [x]_0^2 = 50\pi$
$V = 50\pi$. $V = \pi \int_0^2 25 dx = 25\pi [x]_0^2 = 50\pi$
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What is the role of the function $f(x)$ in the disc method when revolving around the x-axis?
What is the role of the function $f(x)$ in the disc method when revolving around the x-axis?
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Defines the radius of discs. Function determines how far each disc extends from the axis.
Defines the radius of discs. Function determines how far each disc extends from the axis.
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What is the formula for the volume of a solid of revolution using the disc method around the x-axis?
What is the formula for the volume of a solid of revolution using the disc method around the x-axis?
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$V = \int_a^b \pi [f(x)]^2 , dx$. Formula integrates $\pi$ times the radius squared over the interval.
$V = \int_a^b \pi [f(x)]^2 , dx$. Formula integrates $\pi$ times the radius squared over the interval.
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Find the volume of the solid obtained by revolving $x=2$ from $y=0$ to $y=4$ around the y-axis.
Find the volume of the solid obtained by revolving $x=2$ from $y=0$ to $y=4$ around the y-axis.
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$V = 16\pi$. $V = \pi \int_0^4 4 dy = 4\pi [y]_0^4 = 16\pi$
$V = 16\pi$. $V = \pi \int_0^4 4 dy = 4\pi [y]_0^4 = 16\pi$
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Find the volume of the solid obtained by revolving $x=\frac{1}{y}$ from $y=1$ to $y=2$ around the y-axis.
Find the volume of the solid obtained by revolving $x=\frac{1}{y}$ from $y=1$ to $y=2$ around the y-axis.
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$V = \frac{\pi}{2}$. $V = \pi \int_1^2 \frac{1}{y^2} dy = \pi [-\frac{1}{y}]_1^2 = \frac{\pi}{2}$
$V = \frac{\pi}{2}$. $V = \pi \int_1^2 \frac{1}{y^2} dy = \pi [-\frac{1}{y}]_1^2 = \frac{\pi}{2}$
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What is the radius of each disc when revolving $y=f(x)$ around the x-axis?
What is the radius of each disc when revolving $y=f(x)$ around the x-axis?
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$f(x)$. Function value gives distance from x-axis to curve.
$f(x)$. Function value gives distance from x-axis to curve.
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Which axis is used for integration when revolving around the y-axis?
Which axis is used for integration when revolving around the y-axis?
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y-axis. Integration variable matches the axis of revolution.
y-axis. Integration variable matches the axis of revolution.
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Which axis is used for integration when revolving around the x-axis?
Which axis is used for integration when revolving around the x-axis?
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x-axis. Integration variable matches the axis of revolution.
x-axis. Integration variable matches the axis of revolution.
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Identify the variable of integration when revolving around the y-axis using the disc method.
Identify the variable of integration when revolving around the y-axis using the disc method.
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$y$. Integration follows the axis of revolution.
$y$. Integration follows the axis of revolution.
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State the geometric shape formed by the cross-section in the disc method.
State the geometric shape formed by the cross-section in the disc method.
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Circle. Cross-sections perpendicular to the axis are circular discs.
Circle. Cross-sections perpendicular to the axis are circular discs.
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Identify the variable of integration when revolving around the x-axis using the disc method.
Identify the variable of integration when revolving around the x-axis using the disc method.
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$x$. Integration follows the axis of revolution.
$x$. Integration follows the axis of revolution.
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What is the volume formula for a solid of revolution around the y-axis using the disc method?
What is the volume formula for a solid of revolution around the y-axis using the disc method?
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$V = \int_c^d \pi [g(y)]^2 , dy$. Formula integrates $\pi$ times the radius squared along the y-axis.
$V = \int_c^d \pi [g(y)]^2 , dy$. Formula integrates $\pi$ times the radius squared along the y-axis.
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Identify the variable of integration when revolving around the x-axis using the disc method.
Identify the variable of integration when revolving around the x-axis using the disc method.
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$x$. Integration follows the axis of revolution.
$x$. Integration follows the axis of revolution.
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What is the formula for the volume of a solid of revolution using the disc method around the x-axis?
What is the formula for the volume of a solid of revolution using the disc method around the x-axis?
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$V = \int_a^b \pi [f(x)]^2 , dx$. Formula integrates $\pi$ times the radius squared over the interval.
$V = \int_a^b \pi [f(x)]^2 , dx$. Formula integrates $\pi$ times the radius squared over the interval.
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Identify the axis of symmetry for the volume of revolution problem using the disc method.
Identify the axis of symmetry for the volume of revolution problem using the disc method.
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The axis of revolution. Solid rotates around this line creating circular symmetry.
The axis of revolution. Solid rotates around this line creating circular symmetry.
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In the disc method, what shape is formed when revolving a function around an axis?
In the disc method, what shape is formed when revolving a function around an axis?
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Solid of revolution. Revolution creates a 3D solid from the 2D region.
Solid of revolution. Revolution creates a 3D solid from the 2D region.
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What is the cross-sectional area of a disc with radius $r$?
What is the cross-sectional area of a disc with radius $r$?
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$\pi r^2$. Standard formula for area of a circle.
$\pi r^2$. Standard formula for area of a circle.
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Find the volume of the solid obtained by revolving $y=3x$ from $x=0$ to $x=1$ around the x-axis.
Find the volume of the solid obtained by revolving $y=3x$ from $x=0$ to $x=1$ around the x-axis.
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$V = 3\pi$. $V = \pi \int_0^1 9x^2 dx = 3\pi [x^3]_0^1 = 3\pi$
$V = 3\pi$. $V = \pi \int_0^1 9x^2 dx = 3\pi [x^3]_0^1 = 3\pi$
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Given $y=f(x)$, identify the limits of integration when revolving from $x=a$ to $x=b$ around the x-axis.
Given $y=f(x)$, identify the limits of integration when revolving from $x=a$ to $x=b$ around the x-axis.
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$a$ to $b$. Limits match the given x-interval for the region.
$a$ to $b$. Limits match the given x-interval for the region.
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Find the volume of the solid obtained by revolving $y=5$ from $x=0$ to $x=2$ around the x-axis.
Find the volume of the solid obtained by revolving $y=5$ from $x=0$ to $x=2$ around the x-axis.
Tap to reveal answer
$V = 50\pi$. $V = \pi \int_0^2 25 dx = 25\pi [x]_0^2 = 50\pi$
$V = 50\pi$. $V = \pi \int_0^2 25 dx = 25\pi [x]_0^2 = 50\pi$
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What is the role of the function $f(x)$ in the disc method when revolving around the x-axis?
What is the role of the function $f(x)$ in the disc method when revolving around the x-axis?
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Defines the radius of discs. Function determines how far each disc extends from the axis.
Defines the radius of discs. Function determines how far each disc extends from the axis.
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Which axis is used for integration when revolving around the y-axis?
Which axis is used for integration when revolving around the y-axis?
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y-axis. Integration variable matches the axis of revolution.
y-axis. Integration variable matches the axis of revolution.
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Which axis is used for integration when revolving around the x-axis?
Which axis is used for integration when revolving around the x-axis?
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x-axis. Integration variable matches the axis of revolution.
x-axis. Integration variable matches the axis of revolution.
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