- #1
shunae95
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Find the volume of the solid generated by revolving the region enclosed by y= cos x and y = -cos x for [-pi/2, pi/2] about the line y=2pi.
Find Volume of Revolved Region: [-pi/2, pi/2] & y= cos x & y = -cos x
In summary, the volume of the solid generated by revolving the region enclosed by y= cos x and y = -cos x for [-pi/2, pi/2] about the line y=2pi can be found using the washer method with an integral of 16π^2. Alternatively, the shell method can also be used with the same result.
- #2
Greg
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Hello shunae95 and welcome to MHB! :D
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- #3
shunae95
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2π∫π/2 0 (2π+cosx)2−(2π−cosx)2dx
2π∫π/2 (2π+cosx)2−(2π−cosx)2dx16π2∫π/20cosxdx
16π2∫0π/2cosxdx16π2[sinx]π/20=16π2
2π∫π/2 (2π+cosx)2−(2π−cosx)2dx16π2∫π/20cosxdx
16π2∫0π/2cosxdx16π2[sinx]π/20=16π2
- #4
Greg
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shunae95 said:
Do you have to use shells? Disk integration seems much more efficient:
\(\displaystyle \pi\int_{-\pi/2}^{\pi/2}\left(2\pi+\cos(x)\right)^2-\left(2\pi-\cos(x)\right)^2\,dx=16\pi^2\)
My apologies for the late reply.
Last edited:
- #5
MarkFL
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If I use the shell method, I get:
\(\displaystyle V=4\pi\int_{-1}^1 (2\pi-y)\arccos(|y|)\,dy=16\pi^2\)
Greg, when I evaluate the integral you correctly set up using the washer method, I also get $V=16\pi^2$. D
\(\displaystyle V=4\pi\int_{-1}^1 (2\pi-y)\arccos(|y|)\,dy=16\pi^2\)
Greg, when I evaluate the integral you correctly set up using the washer method, I also get $V=16\pi^2$. D
- #6
Greg
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I forgot the coefficients of $2\pi$! 
Thanks Mark! (I've edited my post).
Thanks Mark! (I've edited my post).FAQ: Find Volume of Revolved Region: [-pi/2, pi/2] & y= cos x & y = -cos x
What is the formula for finding the volume of a revolved region?
The formula for finding the volume of a revolved region is V = π∫a^b(y)^2dx, where a and b represent the bounds of the region and y is the function of the curve being revolved around the x-axis.
How do you determine the bounds of the revolved region?
The bounds of the revolved region are determined by the given range of x-values, which in this case is [-pi/2, pi/2]. These values represent the limits of rotation around the x-axis.
What is the difference between y = cos x and y = -cos x?
The difference between these two functions is that y = cos x represents the positive half of the cosine curve, while y = -cos x represents the negative half. This will result in different volumes when revolved around the x-axis.
Can this formula be used for other shapes besides curves?
No, this formula is specifically for finding the volume of a revolved region created by a curve. Other shapes may require different formulas or methods for finding their volume.
Are there any limitations to using this formula?
One limitation of this formula is that it can only be used for revolved regions that are symmetric about the x-axis, since it relies on the integration of (y)^2. Additionally, the function being revolved must also be continuous and differentiable within the given bounds.