find the volume of the solid obtained by revolving the region bounded by the graphs of y=2+4x-x^2 and y=2 about the line y=2.
y-2=4x-x^2
The volume=pi*Int.[0,4](4x-x^2)^2dx=
pi*Int.[0,4](16x^2-8x^3+x^4)dx=
pi*(16/3x^3-2x^4+1/5x^5)[0,4]=
pi*(1024/3-512+1024/5)=pi*1024/30
To find the volume of the solid obtained by revolving the region bounded by the given graphs about the line y=2, we can use the method of cylindrical shells.
First, let's plot the given graphs:
y = 2 + 4x - x^2 (equation 1)
y = 2 (equation 2)
To find the points where the graphs intersect, we set equation 1 and equation 2 equal to each other:
2 + 4x - x^2 = 2
Simplifying the equation, we get:
4x - x^2 = 0
Factoring out an x, we have:
x(4 - x) = 0
So, either x = 0 or 4 - x = 0. This gives us two x-values: x = 0 and x = 4.
Now, let's integrate the circumference of the cylindrical shells to find the volume. The radius of each cylindrical shell is the distance from the line y=2 to the respective function, which is given by:
r = y - 2
We need to express y in terms of x, so rearranging equation 1:
y = 2 + 4x - x^2
The height of each cylindrical shell is the difference in y-values between the two graphs at a given x-value.
h = (2 + 4x - x^2) - 2
Simplifying, we get:
h = 4x - x^2
The volume of each cylindrical shell is given by:
dV = 2πrh dx
Substituting the expressions for r and h, we have:
dV = 2π(4x - x^2)(y - 2) dx
To find the total volume, we integrate this expression with respect to x over the interval x=0 to x=4:
V = ∫(0 to 4) 2π(4x - x^2)(2 + 4x - x^2 - 2) dx
V = ∫(0 to 4) 2π(4x - x^2)(2x - x^2) dx
V = 2π ∫(0 to 4) (4x - x^2)(2x - x^2) dx
Now, we can solve this integrand using standard techniques of integration. After integrating, you will have your final volume.