Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
y=0, y=cos(7x) , x=π/14, x=0 about the axis y=−8
The region is just 1/2 of one arch of the curve
using discs of thickness dx,
v = ∫[0,π/14] π(R^2-r^2) dx
where R = 8+cos7x and r = 8
v = ∫[0,π/14] π((8+cos(7x))^2-8^2) dx = π/28 (π+64)
What about this one:
Find the volume formed by rotating the region enclosed by:
x=5.5y and y^3=x with y≥0 about the y-axis
I'm just confused on which formula I should be using to solve these problems
there are only two formulas.
discs: v = πr^2 dx (or dy)
shells: v = 2πrh dx (or dy)
Sometimes one is more convenient than the other, depending on the boundary of the region. Always sketch the region to get a feel for it.
For this one, either one works.
find where the curves intersect. The region is convex, so having to shift boundaries will not be a problem. There is a problem almost identical to this at
https://www.jiskha.com/questions/1837054/find-the-volume-formed-by-rotating-about-the-y-axis-the-region-enclosed-by-x-11y-and-y-3-x
the answer is kinda messy, though.
could you use either equation for all problems ?
yes, but you may have to split it up into more than one part as you try to avoid duplicating bits. Consider the region bounded by
y = e^x, y = e^-x and the line x=1.
If you rotate that around the x-axis using washers (discs with holes)
v = ∫[0,1] π((e^x)^2 - (e^-x)^2) dx
Using shells, you have to change boundaries at y=1
v = ∫[1/e,1] 2πy(1-ln(1/y)) dy + ∫[1,e] 2πy(1-lny) dy
To rotate around the y-axis, things change. Using shells,
v = ∫[0,1] 2πx(e^x - e^-x) dx
Using washers,
v = ∫[1/e,1] π(1^2 - (ln(1/y))^2) dy + ∫[1,e] π(1^2 - (lny)^2) dy
To find the volume of the solid obtained by rotating the region bounded by the curves about the specified axis, we can use the method of cylindrical shells.
First, let's sketch the curves and solid to visualize the problem. The region bounded by the curves y=0, y=cos(7x), x=π/14, and x=0 is a region in the first quadrant of the coordinate plane. When this region is rotated about the y= -8 axis, it forms a solid with a cylindrical shape.
To use cylindrical shells, we need to express the curves in terms of x. Since we have y=cos(7x), we can solve for x to get x=arccos(y)/7.
Now, let's express the limits of integration in terms of y:
The lower limit of integration, y=0, corresponds to x=arccos(0)/7= π/14.
The upper limit of integration, y=cos(7π/14)=cos(π/2)=0, corresponds to x=arccos(0)/7= π/14.
The height of each cylindrical shell is given by the difference between the y-coordinate of the curve y=cos(7x) and the axis y= -8. So, the height h = cos(7x) - (-8) = cos(7x) + 8.
The radius of each cylindrical shell can be calculated as r = x = π/14.
Now, we can determine the volume of each cylindrical shell by using the formula:
V = 2πrhΔx
To find the total volume, we need to integrate this expression with respect to x from the lower limit to the upper limit:
V = ∫[π/14 to π/14] 2π * x * (cos(7x) + 8) dx
Now you can evaluate this integral to find the volume of the solid.