Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
x = 7y^2, y ≥ 0, x = 7; about y = 2
The region has vertices at
(0,0) (7,0) (7,1)
So, using shells of thickness dy, we have
v = ∫[0,1] 2πrh dy
where r = 2-y and h = 7-x = 7-7y^2
v = ∫[0,1] 2π(2-y)(7-7y^2) dy = 91π/6
Or, using discs (washers) of thickness dx,
v = ∫[0,7] π(R^2-r^2) dx
where R=2 and r=2-y=2-√(x/7)
v = ∫[0,7] 2π(2^2-(2-√(x/7))^2) dx = 91π/6
Well, well, well, it seems like we have a mathematical conundrum on our hands! Let's see if we can crack this one using the method of cylindrical shells.
To find the volume generated by rotating the region bounded by the curves x = 7y^2 and x = 7 about the axis y = 2, we can use cylindrical shells.
First, let's find the limits of integration. To do that, we need to figure out where these curves intersect.
Setting the two equations equal to each other, we have:
7y^2 = 7
Dividing both sides by 7, we get:
y^2 = 1
Taking the square root of both sides, we find y = ±1. However, since we are only considering the region where y ≥ 0, we'll take y = 1 as our upper limit of integration.
Now, let's move on to setting up the integral using cylindrical shells. The volume of each cylindrical shell is given by the formula:
dV = 2πrh * dx
In this case, the radius r is given by: r = 7 - x
And the height h is given by: h = y - 2
We want to express everything in terms of x since we'll be integrating with respect to x.
To eliminate y, we can solve the equation x = 7y^2 for y and find: y = sqrt(x/7)
Substituting these expressions for r and h into the shell volume formula, we get:
dV = 2π(x/7 - 2) * (7 - x) * dx
Now, all that's left is to integrate from x = 0 to x = 7.
V = ∫[0 to 7] (2π(x/7 - 2) * (7 - x)) dx
And there you have it! You can now proceed to calculate the integral to find the volume generated by rotating the region bounded by those curves about the axis y = 2.
I hope this explanation helped you visualize the process! If not, well, I tried my best to entertain you with this mathematical extravaganza. Keep those questions coming, and I'll keep the humor flowing!
To find the volume generated by rotating the region bounded by the curves x = 7y^2, y ≥ 0, and x = 7 about the line y = 2, we can use the method of cylindrical shells.
First, let's graph the region to have a better visual understanding:
Graph:
(7, 4)
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(0,0)
The region bounded by the curves x = 7y^2, y ≥ 0, and x = 7 is a parabolic region extending from the y-axis to the point (7, 4) on the x-axis. When rotated about y = 2, it will generate a solid shape with cylindrical shells.
To find the volume of each cylindrical shell, we need to consider an infinitesimally small strip of width dy along the y-axis.
The radius of each cylindrical shell can be determined by subtracting the x-coordinate of the parabolic curve (x = 7y^2) from the distance between the axis of rotation (y = 2) and the curve.
Considering an infinitesimally small strip of width dy:
Radius = (2 - 7y^2)
Height of the cylindrical shell = dx = dy
Volume of the cylindrical shell = 2π(radius)(height)dy = 2π(2 - 7y^2)dy
To find the total volume generated by rotating the region, we need to integrate this expression from the lower limit of y (0) to the upper limit of y (2) since that's where the region is bounded.
V = ∫[0,2] 2π(2 - 7y^2)dy
Now, we can proceed to evaluate the integral:
V = 2π ∫[0,2] (2 - 7y^2)dy
= 2π [2y - (7/3)y^3] |[0,2]
= 2π [(2*2 - (7/3)*2^3) - (2*0 - (7/3)*0^3)]
= 2π [(4 - (56/3)) - 0]
= 2π [(12/3) - (56/3)]
= 2π [-44/3]
= -88π/3
Therefore, the volume generated by rotating the region bounded by x = 7y^2, y ≥ 0, and x = 7 about y = 2 is -(88π/3) or approximately -92.07 cubic units.
To find the volume generated by rotating the region about the y-axis using the method of cylindrical shells, we need to set up an integral that represents the volume of each individual cylinder.
1. First, let's visualize the region and the axis of rotation. The given curves are x = 7y^2 and x = 7, and we want to rotate this region about the line y = 2.
2. Next, we need to determine the limits of integration. Since the y-values start from 0 and go up to a certain value, we will integrate with respect to y.
3. Let's find the points where the curves intersect. Setting the two equations equal to each other, we have:
7y^2 = 7
y^2 = 1
y = ±1
Since we are given that y ≥ 0, we only consider y = 1.
4. Now, let's set up the integral using cylindrical shells. Each cylinder has a height of dy (infinitesimally small width), and its radius is the distance from the axis of rotation (y = 2) to the curve x = 7y^2. The volume of each cylinder is given by the formula V = 2πrhdy, where r is the radius and h is the height.
5. The radius r is the distance from y = 2 to the curve x = 7y^2. Since the axis of rotation is y = 2, the radius is equal to 7y^2 - 2.
6. The height h is dy, since it represents the width or thickness of each cylindrical shell.
7. The limits of integration for y are from 0 to 1.
8. Putting all these steps together, the integral to find the volume V is:
V = ∫[0,1] 2π(7y^2 - 2)dy
9. Integrate the expression with respect to y using the limits of integration determined in step 7.
10. Finally, evaluate the integral to find the volume V.