The region between the graphs of y=x^2 and y=2x is rotated around the line x=2.
The volume of the resulting solid is ____. Please help. I've tried this problem so many times but i keep getting it wrong. I thought the outer radius would be (2-0) and inner radius is (2-x^2) integrated from 0 to 2, but its not right.
They intersect at (0,0) and (2,4)
Since you are rotationg about x = 2, we need to take horizontal slices and our
general formula will be
vol = pi(integral)(outer radius)^2 - (inner radius)^2 dy from 0 to 4
Vol = pi(integral)((2-y/2)^2 - (2-√y)^2)dy from 0 to 4
= pi[(1/12)y^3 - (3/2)y^2 + (8/3)^(3/2)] from 0 to 4
My answer came out to be 8/3
Let me know if your text agrees.
=
To find the volume of the resulting solid when the region between the graphs of y = x^2 and y = 2x is rotated around the line x = 2, we'll use the method of cylindrical shells.
First, we need to determine the limits of integration. Since we're rotating the region around the line x = 2, we are integrating from x = 0 to x = 2.
Next, we need to consider the infinitesimally small shell that we're rotating around the x-axis. To find the height (or circumference) of this shell, we need to calculate the difference between the curves y = 2x and y = x^2. We subtract the equation of the lower curve (y = x^2) from the equation of the upper curve (y = 2x):
h(x) = (2x) - (x^2) = 2x - x^2
Now let's find the radius of the shell. As the shell is being rotated around the line x = 2, the distance from the axis of rotation to the edge of the shell corresponds to the difference between the x-coordinate and the line x = 2:
r(x) = x - 2
To determine the volume of the shell, we'll use the formula for the volume of a cylindrical shell:
V_shell = 2π * r(x) * h(x) * dx
To find the total volume of the solid, we'll integrate the volume of the shells from x = 0 to x = 2:
V_total = ∫[0,2] (2π * r(x) * h(x)) dx
Now, let's calculate the integral:
V_total = ∫[0,2] (2π * (x - 2) * (2x - x^2)) dx
Expand the equation:
V_total = 2π * ∫[0,2] ((2x^2 - x^3 - 4x + 2x^2)) dx
Simplify:
V_total = 2π * ∫[0,2] (4x^2 - x^3 - 4x) dx
Integrate:
V_total = 2π * ([(4/3)x^3] - [(1/4)x^4] - [2x^2]) | from 0 to 2
Evaluate the integral:
V_total = 2π * [(4/3)(2^3) - (1/4)(2^4) - 2(2^2)] - 2π * [(4/3)(0^3) - (1/4)(0^4) - 2(0^2)]
Simplify further:
V_total = 2π * [(4/3)(8) - (1/4)(16) - 2(4)] - 2π * [(4/3)(0) - (1/4)(0) - 2(0)]
Evaluate the expression:
V_total = 2π * [32/3 - 4 - 8] - 2π * [0 - 0 - 0]
Simplify:
V_total = 2π * [32/3 - 4 - 8]
Evaluate:
V_total = 2π * (32/3 - 12)
V_total = 2π * (8/3)
Finally, we have:
V_total = (16π/3) cubic units
So, the volume of the resulting solid is (16π/3) cubic units.