The area bounded by the curve y = 2x^2-x^3 and line y=0 is rotated around the y-axis. The volume of the resulting structure can be expressed as V = a(pi)/b, where a and b are coprime positive integers. What is the value of a + b?
Using shells:
v = ∫[0,2] 2πrh dx
where r = x and h = y = 2x^2 - x^3
v = 2π∫[0,2] 2x^3 - x^4 dx
piece of cake from here on
16/5=21
To find the volume of the solid formed by rotating the curve around the y-axis, we can use the method of cylindrical shells. This method involves integrating the surface area of all the infinitely thin cylindrical shells that make up the solid.
First, we need to determine the limits of integration for y. The curve y = 2x^2 - x^3 intersects the x-axis when y = 0. So, we need to solve for x when y = 0:
0 = 2x^2 - x^3
0 = x^2(2 - x)
x = 0 or x = 2
Therefore, our limits of integration for y are from y = 0 to y = 2.
Next, we need to express x in terms of y for this curve. Rearranging the equation, we have:
2x^2 - x^3 = y
x^3 - 2x^2 + y = 0
At this point, finding the exact inverse is not straightforward. We can use numerical methods or approximations to solve for x in terms of y. However, since we are only interested in the value of a + b, we will approximate the integral using calculus.
The volume of a cylindrical shell is given by V = 2πrhΔy, where r is the distance from the axis of rotation to the shell (which is x in this case), h is the height of the shell, and Δy is the thickness of the shell.
In this case, r = x, h = 2x^2 - x^3, and Δy = dy since we are integrating with respect to y.
The volume of the solid can be expressed as:
V = ∫[from 0 to 2] 2πx(2x^2 - x^3) dy
Using the approximation, we have:
V ≈ ∑[from 0 to N] 2πx_i(2x_i^2 - x_i^3)Δy_i
where N is the number of partitions, x_i is the value of x at each partition i, and Δy_i is the thickness of each partition.
We can then calculate the sum of these approximate volumes for a range of N values.
Finally, we can find the value of a + b by evaluating the integral or sum, obtaining the value of V, and identifying the coprime positive integers a and b in the expression V = aπ/b.