use the method of cylindrical shells to find the volume generated by rotation the region bounded by x=y^2+1, x=2, about y=-2. Sketch the region, the solid, and a typical shell.
I know that for shells, its 2pix*f(x)*dx. i am having trouble figuring out what the "x" is for the circumference. I believe f(x), which is the height, is 2-(y^2+1), and dx is the thickness. I sketched the region already, i just need help with setting up the integral.
As always, draw a diagram. The parabola has vertex at (1,0) and opens to the right. The line x=2 intersects the parabola at (2,1) and (2,-1).
So, you will be revolving the sort-of-semicircular chink around the line y=-2, which is below the curve.
So, as you say,
v = ∫2πrh dy
because the shells have a horizontal axis, and thickness dy. I can understand your confusion if trying to use dx. See below.
r = y+2 because the axis of revolution is not the x-axis, but 2 units down.
h = 2-x because we're rotating an area cut off by x=2.
v = ∫2π(y+2)(2-(y^2+1)) dy
I'll let you figure the limits of integration.
If you want to use dx, you need to use discs (washers) of thickness dx. For these,
v = ∫π(R^2-r^2) dx
where R = √(x-1) and r = -√(x-1)
I worked it out and found the answer to be 454pi/12. this is correct right? The limits of integration i used were from 1 to 2.
To find the volume generated by rotating the region about the line y = -2 using the method of cylindrical shells, we need to consider a vertical strip of width dx (thickness) that runs from x = 1 to x = 2. This strip will be rotated around the y = -2 line, creating a cylindrical shell.
Let's go through the steps to set up the integral:
1. Determine the height of the shell, which is the difference between the upper and lower y-values of the region at a given x. In this case, f(x) represents the height of the shell.
Given: f(x) = 2 - (y^2 + 1)
2. Express the height function in terms of x. Since we have an equation in terms of y, we need to rewrite it in terms of x using the given equation x = y^2 + 1 as follows:
x = y^2 + 1
y^2 = x - 1
y = √(x - 1)
The height function becomes: f(x) = 2 - (√(x - 1))^2
Simplifying further, f(x) = 2 - x + 1 = 3 - x
3. Determine the radius of the shell, which is the distance from the axis of rotation (y = -2) to the corresponding x-value on the boundary curve. In this case, the radius is: r(x) = 2 - (-2) = 4
4. Determine the width of the shell (dx), which represents the change in x as we move along the x-axis.
5. Set up the integral using the formula for the volume of a cylindrical shell:
V = ∫[a,b] 2πr(x)f(x) dx
In this case, the boundaries for integration are a = 1 and b = 2. The integral becomes:
V = ∫[1,2] 2π(4)(3 - x) dx
6. Simplify and evaluate the integral:
V = 8π ∫[1,2] (3 - x) dx
= 8π [(3x - (x^2/2))] evaluated from 1 to 2
= 8π [(3(2) - (2^2/2))] - [(3(1) - (1^2/2))]
= 8π [6 - 2] - [3 - 1/2]
= 8π [4] - [2.5]
= 31.42
Therefore, the volume generated by rotating the region bounded by x = y^2 + 1, x = 2, about y = -2 is approximately 31.42 cubic units.
To find the volume using the method of cylindrical shells, we need to integrate the product of the circumference, height, and thickness of the shells along the given region.
First, let's sketch the region bounded by the curves x = y^2 + 1, x = 2, and the line y = -2. It is important to visualize the region before proceeding to set up the integral.
The region looks like a parabola opening to the right, starting at (1, -2) and reaching x = 2, with the line y = -2 as the bottom boundary. It will be helpful to label these points on the graph. Also, we can mark a typical shell for better understanding.
Now, let's express the given equations in terms of y to make integration easier:
x = y^2 + 1
Rearranging, we get:
y^2 = x - 1
y = ±√(x - 1)
Here, we can see that the region is bounded by two curves: the upper curve is y = √(x - 1) and the lower curve is y = -√(x - 1).
Next, we need to determine the height of each shell. The height function (f(x)) is the vertical distance between the line y = -2 and the upper curve y = √(x - 1). So:
f(x) = √(x - 1) - (-2)
= √(x - 1) + 2
Now, to find the circumference of each shell, imagine a vertical line intersecting the solid at x = x₀. To calculate the circumference, we need to find the corresponding value of y on the upper curve. Since the upper curve is given by y = √(x - 1), we have:
x = y^2 + 1
x₀ = y₀^2 + 1
Therefore, the circumference is given by 2πy₀.
Finally, the thickness of each shell is dx, which represents an infinitesimally small change in x.
Putting it all together, the integral for finding the volume generated by rotating the region about y = -2 is:
V = ∫[x₁, x₂] 2πy₀ * f(x₀) * dx₀
= ∫[x₁, x₂] 2π(√(x₀ - 1)) * (√(x₀ - 1) + 2) * dx₀
Here, x₁ and x₂ represent the x-coordinates of the points where the curves intersect.
Now, you can substitute the appropriate limits of integration for x₁ and x₂ and evaluate the integral to find the volume.
Remember to evaluate the integral using appropriate anti-derivative techniques or numerical methods.