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?
something's wrong. The region is not closed.
i believe its bounded from y=0 to y=2
Ah. In that case, using discs (washers),
v = ∫[0,2] π(R^2-r^2) dy
where R = 4y and r = 2y^2, so
v = ∫[0,2] π((4y)^2-(2y^2)^2) dy
= 4π∫[0,2] 4y^2 - y^4 dy
= 1024/15 π
no, i think that
∫[0,2] π((4y)^2-(2y^2)^2) dy is 256/15π
you may be right. maybe I mixed in an unneeded factor of 4 somewhere.
To find the volume of the structure, we need to use the method of cylindrical shells. This involves calculating the volume of each infinitesimally thin cylinder and then integrating them to find the total volume.
First, let's find the points of intersection between the curve 2y^2 = x and the line 4y = x. To do this, we can equate the two equations:
2y^2 = x
4y = x
By substituting 4y for x in the first equation, we get:
2y^2 = 4y
Simplifying the equation, we have:
2y^2 - 4y = 0
Factoring out a common factor of 2y, we get:
2y(y - 2) = 0
Setting each factor equal to zero gives us two possible values for y: y = 0 and y = 2.
Now that we have the limits of integration, let's calculate the volume of each cylindrical shell. Each cylindrical shell will have a radius and height given by:
Radius = x = 2y^2
Height = 4y - x = 4y - 2y^2
The volume of each cylindrical shell can be calculated by multiplying the circumference of the shell by its height:
dV = 2π(radius)(height) = 2π(2y^2)(4y - 2y^2) = 4πy^3 - 8πy^4
To find the total volume, we need to integrate this expression with respect to y, from y = 0 to y = 2:
V = ∫(4πy^3 - 8πy^4) dy
Evaluating this integral, we get:
V = π(y^4 - 2y^5) | from 0 to 2
Substituting the limits of integration, we have:
V = π((2^4 - 2(2)^5) - (0^4 - 2(0)^5))
= π(16 - 64) - (0 - 0)
= π(-48)
Therefore, a = -48 and b = 1. Since we are only interested in positive integers, we disregard the negative sign. Hence, a + b = -48 + 1 = -47.
Thus, the value of a + b is -47.