the region in the first quadrant bound by the graphs y=(x*x*x)/3 and y=2x is revolved about the y-axis. find the volume of the resulting solid.
To find the volume of the resulting solid when the region bounded by the graphs y=(x^3)/3 and y=2x in the first quadrant is revolved about the y-axis, we can use the method of cylindrical shells.
First, we need to find the points where the two curves intersect. Setting the two equations equal to each other, we have:
(x^3)/3 = 2x
Multiply both sides by 3 to eliminate the fraction:
x^3 = 6x
Rearranging the equation to one side, we have:
x^3 - 6x = 0
Factor out an x:
x(x^2 - 6) = 0
We have two solutions: x = 0 and x = √6.
Next, we need to determine the limits of integration for the volume integral. Since we are revolving the region about the y-axis, the limits will be from y = 0 to y = 2x.
Now, we can set up the volume integral using the cylindrical shell method:
V = ∫[0 to 2√6] 2πy(x) dx
where y(x) is the height of a cylindrical shell at a given x-value.
Since the shells are formed by rotating the region bounded by y=(x^3)/3 and y=2x, the height of a shell is given by the difference between the y-values of the two curves at a particular x-value:
y(x) = 2x - (x^3)/3
Now, we can simplify the integral:
V = 2π ∫[0 to 2√6] (2x - (x^3)/3) dx
Evaluating the integral will give you the volume of the resulting solid.