The area bounded by the curve 2y^2=x and the line 4y=x is rotated around the y-axis. The volume of the resulting structure can be expressed as V=a/bπ, where a and b are coprime positive integers. What is the value of a+b?
To find the volume of the structure formed by rotating the area bounded by the given curve and line around the y-axis, we can use the method of cylindrical shells.
First, let's find the points of intersection between the curve 2y^2 = x and the line 4y = x. We can equate the two equations:
2y^2 = x
4y = x
Setting them equal to each other:
2y^2 = 4y
Rearranging the equation:
2y^2 - 4y = 0
Factoring out 2y:
2y(y - 2) = 0
We have two solutions:
y = 0 and y = 2
Since y = 0 is not in the desired range, we'll consider the interval y ∈ [0, 2] for rotation.
Next, we will find the radius and height for each cylindrical shell. The radius (r) will be the distance from the y-axis to the curve 2y^2 = x. Since we are rotating around the y-axis, the radius is simply x.
Since x = 2y^2, the radius becomes r = 2y^2.
The height (h) of each cylindrical shell will be the distance between two adjacent points on the curve. Considering a small change in y, we can express h as dh.
The differential volume (dV) of each cylindrical shell can be calculated as follows:
dV = 2πrh dh
= 2π(2y^2) dh
= 4πy^2 dh
To find the total volume (V), we integrate the differential volume over the range of y from 0 to 2:
V = ∫[0 to 2] 4πy^2 dh
Integrating with respect to y:
V = 4π ∫[0 to 2] y^2 dh
= 4π [(1/3) y^3] [0 to 2]
= 4π [(1/3) (2)^3 - (1/3) (0)^3]
= 4π (8/3)
= (32/3)π
Therefore, the value of a is 32 and b is 3. The sum a + b is 32 + 3 = 35.
Hence, the value of a + b is 35.