region bounded by the parabolas y=x^2 and y=6x-(x^2) is rotated about the x-axis so that a vertical line segment cut off by the curves generates a ring. find the value of x for which we obtain the ring of largest area
To find the value of x for which we obtain the ring of largest area, we need to set up the integral that represents the area of the ring.
First, let's find the points of intersection between the two parabolas. We set the two equations equal to each other:
x^2 = 6x - x^2
2x^2 - 6x = 0
Factor out 2x:
2x(x - 3) = 0
This equation gives us two solutions: x = 0 and x = 3. Therefore, the ring is formed between x = 0 and x = 3.
To set up the integral for finding the area, we need to consider the hollow disk formed by rotating a vertical line segment about the x-axis. The area of this hollow disk can be calculated by taking the difference between the areas of two disks.
The area of an individual disk is given by A = πr^2, where r is the radius of the disk. In this case, since we are rotating around the x-axis, the radius of the outer disk is given by the function y = 6x - x^2, and the radius of the inner disk is given by the function y = x^2.
The outer radius is (6x - x^2), and the inner radius is (x^2). Therefore, the area of the ring (A_ring) can be calculated as:
A_ring = π(outer radius)^2 - π(inner radius)^2
A_ring = π((6x - x^2)^2 - (x^2)^2)
Simplifying this expression, we get:
A_ring = π(36x^2 - 12x^3 + x^4 - x^4)
A_ring = π(36x^2 - 12x^3)
Now, we need to find the value of x that maximizes this area. To do this, we can take the derivative of the area with respect to x and set it equal to zero:
dA_ring/dx = 72x - 36x^2
Setting the derivative equal to zero and solving for x:
72x - 36x^2 = 0
Factor out 36x:
36x(2 - x) = 0
This equation gives us two solutions: x = 0 and x = 2. However, since we are looking for the value of x within the range of x = 0 to x = 3, we can conclude that the value of x for which we obtain the ring of largest area is x = 2.
Thus, the value of x for which we obtain the ring of largest area is x = 2.