Find the volume of the solid obtained by rotating the region enclosed by y=x^2, y=6x about the line x=0using the method of disks or washers.
How did u get all of that
the curves intersect at (0,0) and (6,36)
using washers of thickness dx, we have
v = ∫[0,6] π(R^2-r^2) dx
where R=6x and r=x^2
v = ∫[0,6] π(36x^2-x^4) dx = 5184π/5
To check, try using shells of thickness dy:
v = ∫[0,36] 2πrh dy
where r=y and h=√y-y/6
v = ∫[0,36] 2πy(√y-y/6) dy = 5184π/5
Well, isn't this a rotating party! Let's calculate the volume using the method of disks or washers!
First, we need to figure out the limits of integration. To find the volume of the solid obtained by rotating the region enclosed by y=x^2 and y=6x about the line x=0, we need to find where these two functions intersect.
Setting x^2=6x, we can rearrange it to x^2-6x=0. Factoring out an x, we get x(x-6)=0. So, x=0 or x=6.
Now we can set up the integral. We'll integrate from 0 to 6, since those are our limits of integration. The radius of each disk or washer will be the distance from the axis of rotation (x=0) to the function y=x^2 or y=6x, whichever is farther from the rotation axis at any given value of x.
To determine the radius function, we need to find the distance from x=0 to each curve. For the curve y=x^2, this distance is simply x. For the curve y=6x, the distance is 6x.
Since the curve y=6x is farther from the rotation axis at any given x, we'll use it as our radius function.
The area of each disk or washer is given by A=π(r^2), where r is the radius of the disk or washer.
Plugging in our radius function, we get A=π(6x)^2=36πx^2.
Finally, we can integrate the area function from 0 to 6:
Volume = ∫(0 to 6) 36πx^2 dx
Integrating, we get:
Volume = (36π/3)(6^3-0^3)
Volume = 432π cubic units!
So, the volume of the solid obtained by rotating the region enclosed by y=x^2 and y=6x about the line x=0 using the method of disks or washers is 432π cubic units. Let's get this party spinning!
To find the volume of the solid obtained by rotating the region enclosed by y=x^2, y=6x about the line x=0 using the method of disks or washers, we need to integrate the cross-sectional area of the region with respect to x and sum up the volumes.
First, let's find the points of intersection of the two curves by setting them equal to each other:
x^2 = 6x
Rearrange the equation:
x^2 - 6x = 0
Factor out x:
x(x - 6) = 0
Set each factor equal to zero:
x = 0 or x - 6 = 0
Solving for x, we find two points of intersection: x = 0 and x = 6.
Next, we need to set up the integral to find the volume. We will integrate the cross-sectional area from x = 0 to x = 6.
For the method of disks, we consider a thin disk or cylinder of height dy at a particular y value.
The cross-sectional area of the disk at height y is given by A = πr^2, where r is the radius of the disk.
In this case, the radius r is the distance from the axis of rotation (x = 0) to the curve y = x^2 or y = 6x.
At a given y value, the distance from the axis of rotation to the curve y = x^2 is x = sqrt(y), and the distance to the curve y = 6x is x = y/6.
So, the radius r is given by r = y/6 - sqrt(y).
The volume of a disk with height dy is given by dV = A*dy = π(r^2)*dy.
Integrating this expression from y = 0 to y = 36 (the upper limit of the region), we get the volume of the solid as:
V = ∫[0 to 36] π ( (y/6 - sqrt(y))^2 ) dy
Evaluating this integral will give us the volume of the solid obtained by rotating the region enclosed by y = x^2, y = 6x about the line x = 0 using the method of disks or washers.
To find the volume of the solid obtained by rotating the region enclosed by y=x^2, y=6x about the line x=0 using the method of disks or washers, we can follow these steps:
1. Graph the given functions, y=x^2 and y=6x, to visualize the region enclosed between them.
2. Determine the points of intersection of the two functions by setting them equal to each other:
x^2 = 6x
x^2 - 6x = 0
Factor out x: x(x - 6) = 0
So, x = 0 or x = 6.
Therefore, the region of interest is bounded by x = 0 on the left and x = 6 on the right.
3. Set up the integral to find the volume using the method of disks. The volume element of a disk with radius r and thickness dx is given by dV = π * r^2 * dx.
Since we are rotating the region about the line x = 0, the radius of each disk will be x. And the thickness dx will represent an infinitely thin slice of the region.
So, the volume element becomes dV = π * (x^2)^2 * dx = π * x^4 * dx.
4. Integrate the volume element over the range of x from 0 to 6 to find the total volume:
V = ∫[0,6] π * x^4 dx.
Integrating this expression will give you the volume of the solid of revolution.
Note: If you choose to use the method of washers, you will need to set up the integral using the formula dV = π * (Outer radius^2 - Inner radius^2) * dx instead, and the rest of the steps remain the same.