The volume of the solid obtained by rotating the region enclosed by
y=x^2 y=2x
about the line x=2 can be computed using the method of disks or washers via an integral
The volume is V
Find the volume of the solid obtained by rotating the region bounded by
y=5x+25 and y=0
about the y-axis.
To find the volume of the solid obtained by rotating the region enclosed by the curves y = x^2 and y = 2x about the line x = 2, we can use the method of disks or washers via an integral.
First, let's sketch the region enclosed by the curves:
y = x^2 (parabola)
y = 2x (line)
To find the points of intersection of these curves, we can set them equal to each other:
x^2 = 2x
Rearranging this equation:
x^2 - 2x = 0
Factoring out x:
x(x - 2) = 0
This equation yields two solutions: x = 0 and x = 2. However, we are only interested in the region between x = 0 and x = 2, as stated in the problem.
Now, let's consider a vertical slice of this region at a particular value of x, and rotate it about the line x = 2. This will create a disk (or washer) with radius equal to the y-coordinate of the curve at that particular x-value.
The radius of the disk for a given x-value within the interval [0, 2] can be found by subtracting the y-coordinate of the line from the y-coordinate of the parabola:
r = x^2 - (2x)
The differential volume element, dV, of the disk is then given by:
dV = π * r^2 * dx
We integrate this expression from x = 0 to x = 2 to find the total volume, V:
V = ∫[0,2] π * r^2 * dx
Substituting the expression for r, we have:
V = ∫[0,2] π * (x^2 - 2x)^2 dx
Evaluating this integral will give you the volume V of the solid obtained by rotating the region enclosed by the curves y = x^2 and y = 2x about the line x = 2.