Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y-axis.
Y = 3/X, Y = 0, X =5, X = 2
To find the volume using the method of cylindrical shells, we need to integrate the shell volume formula over the given region.
First, let's sketch the region bounded by the curves Y = 3/X, Y = 0, X = 5, X = 2.
The region looks like a trapezoidal region with its two sides extending from X = 2 to X = 5 and its top boundary formed by the curve Y = 3/X. The base of the trapezoid is formed by the X-axis.
Now, let's set up the integral in terms of X.
The height of each shell is given by Y = 3/X.
The radius of each shell is equal to the distance from the y-axis to the curve, which is X.
The shell volume formula is given by V = 2πrh*dx, where r is the radius, h is the height, and dx is the width of the shell.
To calculate the volume, we need to integrate the shell volume formula over the interval from X = 2 to X = 5.
The integral expression for the volume is:
V = ∫[2 to 5] 2π(X)(3/X)dx
Simplifying the expression:
V = 2π ∫[2 to 5] 3 dx
V = 6π ∫[2 to 5] dx
Integrating:
V = 6π(x)|[2 to 5]
V = 6π(5 - 2)
V = 18π
Therefore, the volume generated by rotating the region bounded by the curves about the y-axis is 18π cubic units.