Find the volume using the shell method - about the x-axis
x=y^2/3
v = ∫2πrh dy
where r=y and h=x
v = 2π∫xy dy
Can't tell whether you mean (y^2)/3 or y^(2/3)
anyway, plug it in and use the appropriate limits.
It is y^(2/3) - we have not gone over shell method yet so.....
so, you have
v = 2π∫(y^(2/3))y dy
= 2π∫y^(5/3) dy
= 2π (3/7) y^(7/3)
but you need to specify the limits of integration, since the curve is open-ended.
visit wolframalpha.com and enter
plot y^2 = x^3
The limits are 0,3
To find the volume using the shell method about the x-axis, you need to set up an integral that represents the volume of the solid of revolution.
Given the equation x = y^(2/3), we can rewrite it as y = x^(3/2). Let's consider the region bounded by the curve y = x^(3/2), the x-axis, and the vertical lines x = a and x = b.
First, we need to find the limits of integration, a and b, which represent the x-values where the region starts and ends. To determine these limits, we set y = 0 (equivalent to x = 0) and solve for x^(3/2):
0 = x^(3/2)
x = 0
So the region starts at x = 0.
Next, we need to find the x-value where the curve y = x^(3/2) intersects the x-axis (y = 0) to determine the end of the region. Setting y = 0, we have:
0 = x^(3/2)
x = 0
Since the region ends at x = 0 and starts at x = 0, the limits of integration for our integral are a = 0 and b = 0.
Now, we are ready to set up the integral for the volume using the shell method. The formula for the volume of a shell is V = 2πrh*dx, where r is the distance between the axis of rotation (in this case, the x-axis) and the shell, and h represents the height of the shell.
In our case, the radius r is equal to x, and the height h is equal to y (y = x^(3/2)). The width of the shell, dx, represents an infinitesimally small change in x. Therefore, our integral becomes:
V = ∫(2πxy) dx
Using the limits of integration a = 0 and b = 0, the integral becomes:
V = ∫(0 to 0) (2πxy) dx
Since the limits of integration are the same, the integral evaluates to zero. Hence, the volume of the solid of revolution about the x-axis in this case is zero.