Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 5x^2, y = 5x, x ≥ 0; about the x-axis
V = ???
Sketch the region
Given the region shown at
http://www.wolframalpha.com/input/?i=plot+y%3D5x^2%2Cy%3D5x
and recalling that the volume of a washer with thickness dx is
pi (R^2-r^2) dx
where R=5x and r = 5x^2
it should be clear.
You can verify your answer using shells, where the volume of each shell is
2pi r h dy
where r = y and h = √(y/5) - y/5
To find the volume V of the solid obtained by rotating the region bounded by the curves y = 5x^2 and y = 5x about the x-axis, we need to use the method of cylindrical shells.
First, let's sketch the region bounded by the curves:
- The curve y = 5x^2 is a parabola that opens upward.
- The curve y = 5x is a straight line.
It is important to note that the curves intersect at (0,0) and (1,5).
Here is a rough sketch of the region:
/\
/ \
/ \
----- -----
Now, to find the volume V, we need to integrate the area of the cylindrical shells from x = 0 to x = 1, and multiply by the height of each shell.
The radius of each shell is x, and the height is the difference between the y-values of the two curves at that x-value (y = 5x^2 - 5x).
The volume V can be calculated using the following integral:
V = ∫[from 0 to 1] (2πx)(5x^2 - 5x) dx
Now we can evaluate this integral to find the volume V.
To find the volume of the solid obtained by rotating the region bounded by the curves about the x-axis, we follow these steps:
1. Sketch the region:
First, we need to sketch the region bounded by the curves y = 5x^2 and y = 5x. This region lies in the first quadrant and is bounded by the x-axis, the two curves, and the line x = 0. It is an irregular shape resembling a triangle.
2. Determine the limits of integration:
To find the limits of integration, we need to determine the points where the curves intersect. Setting the two curves equal to each other, we get:
5x^2 = 5x
Dividing both sides by 5 gives:
x^2 = x
Rearranging the equation:
x^2 - x = 0
Factoring out x:
x(x - 1) = 0
This equation is satisfied when x = 0 and x = 1. Therefore, our limits of integration will be from x = 0 to x = 1.
3. Set up the integral:
The general formula for finding the volume of a solid obtained by rotating the region bounded by two curves about the x-axis is:
V = ∫[a,b] π(R^2 - r^2) dx
In this case, the outer radius R is determined by the curve y = 5x^2, and the inner radius r is determined by the curve y = 5x.
So, the integral for finding the volume becomes:
V = ∫[0,1] π((5x^2)^2 - (5x)^2) dx
Simplifying the equation further:
V = ∫[0,1] π(25x^4 - 25x^2) dx
4. Evaluate the integral:
Integrating the equation will give us the volume:
V = π ∫[0,1] (25x^4 - 25x^2) dx
The integral of 25x^4 is (25/5)x^5 = 5x^5/5 = x^5, and the integral of 25x^2 is (25/3)x^3 = 25x^3/3.
So, the volume becomes:
V = π [x^5 - x^3/3] evaluated from 0 to 1
Substituting the limits of integration:
V = π [(1^5 - 1^3/3) - (0^5 - 0^3/3)]
Simplifying the equation further:
V = π [(1 - 1/3) - (0 - 0/3)]
V = π (2/3)
Finally, the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis is 2π/3.