Use the disk method to find the volume of the solid formed by rotating the region bounded by y=2x and y=x^2 about the y-axis.
To use the disk method to find the volume of the solid formed by rotating the region bounded by y = 2x and y = x^2 about the y-axis, follow these steps:
Step 1: Identify the interval of rotation:
We need to determine the points of intersection of the two curves y = 2x and y = x^2. To find this, set the two equations equal to each other:
2x = x^2
0 = x^2 - 2x
0 = x(x - 2)
x = 0 or x = 2
So the interval of rotation is from x = 0 to x = 2.
Step 2: Determine the radius at each point along the interval:
Since we are rotating the region about the y-axis, the radius at each y-value is simply the x-value. So, the radius is x.
Step 3: Write the volume formula:
The volume of each disk is given by the formula V = πr^2h, where r is the radius and h is the thickness of each disk.
Step 4: Determine the thickness of each disk:
The thickness of each disk is represented by Δy, which is the difference in y-values between the curves. In this case, Δy = y(upper) - y(lower).
Δy = (2x) - (x^2) = 2x - x^2
Step 5: Set up the integral to find the volume:
The integral to find the volume is given by V = ∫ πr^2Δy. Substituting the values of r and Δy, we get V = ∫ π(x^2)(2x - x^2) dx, from x = 0 to x = 2.
Step 6: Evaluate the integral:
Integrate the function π(x^2)(2x - x^2) with respect to x from 0 to 2. This will give you the volume of the solid.
V = ∫[0 to 2] π(x^2)(2x - x^2) dx
Now, solve this integral to find the volume.
Once you evaluate the above integral, you will obtain the volume of the solid formed by rotating the given region about the y-axis using the disk method.
To find the volume of the solid formed by rotating the region bounded by the curves y = 2x and y = x^2 about the y-axis using the disk method, we will integrate the cross-sectional areas of the infinitesimally thin disks.
Step 1: Determine the limits of integration
To use the disk method, we need to find the points of intersection between the two curves. Equating the two equations gives us:
2x = x^2
Rearranging the equation, we get:
x^2 - 2x = 0
Factoring out an x, we have:
x(x - 2) = 0
This equation gives us two values of x: x = 0 and x = 2. These will be our limits of integration.
Step 2: Set up the integral for the volume
The volume of each infinitesimally thin disk can be calculated as π * radius^2 * height. In this case, the radius of each disk is given by the x-coordinate of the function y = 2x, and the height is given by the difference between the functions y = 2x and y = x^2.
The radius is x, and the height is the difference between the two functions: 2x - x^2.
The integral for the volume is given by:
V = ∫[a,b] π * (radius)^2 * (height) dx
In this case, a = 0 and b = 2, so the integral becomes:
V = ∫[0,2] π * x^2 * (2x - x^2) dx
Step 3: Evaluate the integral
Now we integrate the expression with respect to x:
V = π * ∫[0,2] (2x^3 - x^4) dx
Integrating each term:
V = π * [(1/2) * x^4 - (1/5) * x^5] evaluated from 0 to 2
V = π * [(1/2) * (2^4) - (1/5) * (2^5)] - [(1/2) * (0^4) - (1/5) * (0^5)]
V = π * [(16/2) - (32/5) - 0]
V = π * [(8) - (32/5)]
V = π * (40/5 - 32/5)
V = π * (8/5)
Finally, simplifying the expression, we get:
V = (8/5)π cubic units
Therefore, the volume of the solid formed by rotating the region bounded by y = 2x and y = x^2 about the y-axis using the disk method is (8/5)π cubic units.