Consider the graph of y2 = x(4 − x)2 (see figure). Find the volumes of the solids that are generated when the loop of this graph is revolved about each of the following
To find the volumes of the solids generated when a loop of the graph is revolved about different axes, we can use the method of cylindrical shells or the disk method, depending on the shape formed.
1. Revolving the loop about the x-axis:
To find the volume using the cylindrical shell method, we divide the loop into infinitesimally small cylindrical shells with radius x and height Δy. We consider a small vertical strip of the loop at x, which corresponds to a shell with radius x and height Δy. The volume of this shell is approximately 2πxΔy times the average height of the strip.
Since we have the equation y^2 = x(4 - x)^2, we can rewrite it as y = (x(4 - x)^2)^(1/2) = (x(4 - x))^(1/2)(2 - x), where y is positive.
To find the boundaries of integration for x, we set y = 0:
0 = (x(4 - x))^(1/2)(2 - x)
Either x = 0 or (4 - x)(2 - x) = 0
Solving the quadratic equation (4 - x)(2 - x) = 0, we get x = 0, x = 2, and x = 4.
Therefore, the volume can be calculated as follows:
V = ∫(x=0 to x=2) 2πx(√(x(4 - x))(2 - x)) dx + ∫(x=2 to x=4) 2πx(-√(x(4 - x))(2 - x)) dx
2. Revolving the loop about the y-axis:
To find the volume using the disk method, we divide the loop into infinitesimally small discs with radius Δx and height y. Each disc has a volume of πy^2Δx.
To express y in terms of x, we need to solve the equation y^2 = x(4 - x)^2 for x in terms of y:
y^2 = x(4 - x)^2
(x(4 - x))^(1/2) = (x(2 - x))(1/2)(2 - x) = y
x = 2 - y^2 / (2 - y)
Now, we can calculate the volume as follows:
V = ∫(y=0 to y=2) π[(2 - y^2 / (2 - y))^2] dy
Note: The integrals in both cases need to be evaluated using appropriate integral techniques, such as u-substitution or other methods.
By following the above steps, you can find the volumes of the solids generated by revolving the loop of the given graph about the x-axis and y-axis.