Find the volume of the solid generated by revolving the region between the y-axis and the curve x = 2/y, 1 ≤ y ≤ 4, about the y-axis.

Steps to Solve

1. Identify the function and limits of integration

The curve is given by ( x = \frac{2}{y} ), which represents the radius of the solid when revolving around the y-axis. The limits of integration are from ( y = 1 ) to ( y = 4 ).

2. Set up the volume integral

To find the volume ( V ) of the solid formed by revolving the area around the y-axis, we use the integral for the volume of revolution:

$$ V = \pi \int_{a}^{b} [f(y)]^2 , dy $$

In our case, ( f(y) = \frac{2}{y} ), therefore,

$$ V = \pi \int_{1}^{4} \left(\frac{2}{y}\right)^2 , dy $$

3. Simplify and compute the integral

First, simplify ( \left(\frac{2}{y}\right)^2 ):

$$ \left(\frac{2}{y}\right)^2 = \frac{4}{y^2} $$

Now substitute this back into the volume integral:

$$ V = \pi \int_{1}^{4} \frac{4}{y^2} , dy $$

4. Evaluate the integral

To evaluate ( \int \frac{4}{y^2} , dy ), we can rewrite it as:

$$ \int \frac{4}{y^2} , dy = 4 \int y^{-2} , dy $$

The integral of ( y^{-2} ) is:

$$ -\frac{4}{y} $$

Thus,

$$ V = \pi \left[-\frac{4}{y}\right]_{1}^{4} $$

Now substitute the limits:

$$ V = \pi \left( -\frac{4}{4} + \frac{4}{1} \right) $$

5. Simplify to find the final volume

Calculating the expression gives us:

$$ V = \pi \left( -1 + 4 \right) = \pi \cdot 3 = 3\pi $$