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
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 find the points of intersection of the two curves. Setting them equal to each other, we have:
x^2 = 2x
Rearranging the equation, we get:
x^2 - 2x = 0
Factoring out an x, we can rewrite it as:
x(x - 2) = 0
So, x = 0 and x = 2 are the points of intersection.
Next, let's consider a vertical strip within the region of rotation. If we rotate this strip about the line x = 2, it will create a disk or washer. The radius of this disk or washer will be the distance from the strip to the line x = 2.
At any point x within the region, the distance from x to the line x = 2 is 2 - x. So the radius of the disk or washer is given by r = 2 - x.
The thickness of the disk or washer is dx, which is an infinitesimally small value.
Now, to find the volume of this disk or washer, we can use the formula:
dV = π * r^2 * dx
Substituting the value of r, we get:
dV = π * (2 - x)^2 * dx
To find the total volume, we integrate this equation from x = 0 to x = 2:
V = ∫[0,2] π * (2 - x)^2 * dx
Evaluating this integral will give us the volume V of the solid.
After evaluating the integral, if you need the exact value of V, you can leave it in terms of π. If you want a numerical approximation, you can substitute the values of π and evaluate the integral numerically using a calculator or computer software.