Find the volume of the solid generated by revolving the region bounded by the graphs of the equations y = 0.5x^2, y = 2, and x = 0 about the line y = 2.
A. 8pi/15
B. 32pi/15
C. 64pi/15
D. 4pi/15
E. 16pi/15
v = ∫ πr^2 dx
where r=2-y
v = ∫[0,2] π(2-x^2/2)^2 dx
or
v = ∫ 2πrh dy
where r=2-y and h=x
v = ∫[0,2] 2π(2-y)√(2y) dy
How do you know the r is 2-y and not just y?
Because it is rotated about the line y = 2 ,
not y = 0 , which would have been the x-axis
To find the volume of the solid generated by revolving the region bounded by the given graphs, we can use the method of cylindrical shells.
First, let's visualize the region bounded by the curves. The equation y = 0.5x^2 represents a parabola that opens upward, the equation y = 2 represents a horizontal line, and the equation x = 0 is the y-axis. So the region looks like a segment of the parabola above the line y = 2, as shown below:
/
/|
/ |
/ |
/___|______
|
|
|
|
|
| y = 2
Next, we need to find the limits of integration. To do this, we need to find the x-values where the curves intersect.
Setting y = 0.5x^2 and y = 2 equal to each other:
0.5x^2 = 2
x^2 = 4
x = ±2
Since we are revolving the region about the line y = 2, we need to find the y-values where the curves intersect with y = 2.
Setting y = 2 in the equation 0.5x^2 = 2:
0.5x^2 = 2
x^2 = 4
x = ±2
So the region is bounded by x = -2, x = 2, y = 0.5x^2, and y = 2.
Now, let's set up the integral to find the volume using the method of cylindrical shells.
The radius of each cylindrical shell is the distance between the line x = 0 and the curve y = 0.5x^2:
radius = x
The height of each cylindrical shell is the difference in y-values between the curves y = 0.5x^2 and y = 2:
height = (0.5x^2 - 2)
The differential volume of each cylindrical shell can be calculated as:
dV = 2π * radius * height * dx
The integral to find the volume becomes:
V = ∫[x=-2 to x=2] 2πx(0.5x^2 - 2) dx
To evaluate this integral, expand the expression inside the integral:
V = ∫[x=-2 to x=2] πx^3 - 4πx dx
Now we can integrate term by term:
V = π * [1/4 * x^4 - 4/2 * x^2] from x=-2 to x=2
V = π * [1/4 * (2)^4 - 4/2 * (2)^2] - [1/4 * (-2)^4 - 4/2 * (-2)^2]
V = π * [1/4 * 16 - 4/2 * 4] - [1/4 * 16 - 4/2 * 4]
V = π * [4 - 8] - [4 - 8]
V = π * (-4) - (-4)
V = -4π + 4π
V = 0
Since the volume is 0, none of the given answer choices match. It is possible that there's an error in either the problem statement or the given answer choices. Can you please recheck the problem or the answer choices?