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DarK1
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x = 2y - y2
x = y
x = y
(Integration) Find volume using disk method when rotated about the X-axis
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- Disk Integration Method Volume
In summary, the disk method gives a volume of twice the radius of the washer, while the washer method gives a volume of three times the radius of the washer.
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MarkFL
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Re: finding VOLUME using integration (DISK METHOD)
The first thing I would do is sketch the region to be revolved:
View attachment 7885
Now, what is the axis of rotation?
The first thing I would do is sketch the region to be revolved:
View attachment 7885
Now, what is the axis of rotation?
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MarkFL
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I didn't realize you had posted the same problem twice, and in the second one I saw, gave the axis of rotation. I have since merged the two threads.
Okay, so we have the axis of rotation as the $x$-axis. Can you identify the following:
Okay, so we have the axis of rotation as the $x$-axis. Can you identify the following:
- The outer radius
- The inner radius
- The limits of integration
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MarkFL
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To follow up, the washer method gives us:
\(\displaystyle V=\pi\int_{x_1}^{x^2} R^2-r^2\,dx\)
The volume of an arbitrary washer is:
\(\displaystyle dV=\pi\left(R^2-r^2\right)\,dx\)
The outer radius is the line $y=x$, and so:
\(\displaystyle R^2=x^2\)
The inner radius is the parabolic curve, but we are given this curve in the form $x(y)$, and we need it in the form $y(x)$. So, we may write it in the form:
\(\displaystyle y^2-2y+x=0\)
We see the axis of symmetry is $y=-\dfrac{-2}{2(1)}=1$, and so we want the lower half of the curve.
Apply the quadratic formula, and discard the larger root:
\(\displaystyle y=\frac{2-2\sqrt{1-x}}{2}=1-\sqrt{1-x}\)
Thus:
\(\displaystyle r^2=(1-\sqrt{1-x})^2=1-2\sqrt{1-x}+1-x=2-2\sqrt{1-x}-x\)
Hence:
\(\displaystyle dV=\pi\left(x^2+x+2\sqrt{1-x}-2\right)\,dx\)
To determine the limits of integration, we equate the two curves:
\(\displaystyle x=1-\sqrt{1-x}\)
Let $u=1-x$, and we have:
\(\displaystyle u=\sqrt{u}\implies u\in\{0,1\}\implies x\in\{0,1\}\)
Hence, adding the slices by integration, we obtain:
\(\displaystyle V=\pi\int_0^1 x^2+x+2\sqrt{1-x}-2\,dx=\frac{\pi}{6}\)
Let's check our result using the shell method:
\(\displaystyle V=2\pi\int_0^1 y\left(\left(2y-y^2\right)-y\right)\,dy=2\pi\int_0^1 y^2-y^3\,dy=\frac{\pi}{6}\quad\checkmark\)
\(\displaystyle V=\pi\int_{x_1}^{x^2} R^2-r^2\,dx\)
The volume of an arbitrary washer is:
\(\displaystyle dV=\pi\left(R^2-r^2\right)\,dx\)
The outer radius is the line $y=x$, and so:
\(\displaystyle R^2=x^2\)
The inner radius is the parabolic curve, but we are given this curve in the form $x(y)$, and we need it in the form $y(x)$. So, we may write it in the form:
\(\displaystyle y^2-2y+x=0\)
We see the axis of symmetry is $y=-\dfrac{-2}{2(1)}=1$, and so we want the lower half of the curve.
Apply the quadratic formula, and discard the larger root:
\(\displaystyle y=\frac{2-2\sqrt{1-x}}{2}=1-\sqrt{1-x}\)
Thus:
\(\displaystyle r^2=(1-\sqrt{1-x})^2=1-2\sqrt{1-x}+1-x=2-2\sqrt{1-x}-x\)
Hence:
\(\displaystyle dV=\pi\left(x^2+x+2\sqrt{1-x}-2\right)\,dx\)
To determine the limits of integration, we equate the two curves:
\(\displaystyle x=1-\sqrt{1-x}\)
Let $u=1-x$, and we have:
\(\displaystyle u=\sqrt{u}\implies u\in\{0,1\}\implies x\in\{0,1\}\)
Hence, adding the slices by integration, we obtain:
\(\displaystyle V=\pi\int_0^1 x^2+x+2\sqrt{1-x}-2\,dx=\frac{\pi}{6}\)
Let's check our result using the shell method:
\(\displaystyle V=2\pi\int_0^1 y\left(\left(2y-y^2\right)-y\right)\,dy=2\pi\int_0^1 y^2-y^3\,dy=\frac{\pi}{6}\quad\checkmark\)
FAQ: (Integration) Find volume using disk method when rotated about the X-axis
What is the disk method for finding volume when rotated about the X-axis?
The disk method is a mathematical technique used to find the volume of a solid created by rotating a two-dimensional shape around a given axis, in this case the X-axis. The volume is calculated by summing the cross-sectional areas of the shape at different points along the axis.
How does the disk method differ from other methods for finding volume?
The disk method is specifically used for finding the volume of solids of revolution, where a two-dimensional shape is rotated around an axis to create a three-dimensional object. Other methods, such as the shell method or washer method, are used for finding the volume of more complex shapes.
What is the formula for using the disk method to find volume when rotated about the X-axis?
The formula for using the disk method to find volume when rotated about the X-axis is V = π ∫ab (f(x))2 dx, where a and b represent the limits of integration and f(x) is the function describing the shape being rotated.
Can the disk method be used to find volume for any shape?
No, the disk method can only be used for finding the volume of solids of revolution, where the shape being rotated is a simple curve (such as a circle or parabola) and the axis of rotation is perpendicular to the direction of integration.
What are some real-world applications of using the disk method to find volume when rotated about the X-axis?
The disk method has many practical applications in fields such as engineering, architecture, and physics. For example, it can be used to calculate the volume of a water tank, the amount of material needed for a cylindrical pipe, or the moment of inertia of a rotating object.