Question

The region in the first quadrant that is bounded above by the curve \(y=1 / \sqrt{x},\) on the left by the line \(x=1 / 4,\) and below by the line \(y=1\) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid by (a) the washer method and (b) the cylindrical shell method.

Short answer

The volume of the solid is \(3\pi\) units^3, whether computed by the washer method or the cylindrical shell method.

Step-by-step solution

Washer Method

For the washer method, set up the integral for volume as follows: \[V = \pi \int_{1/4}^{1} ((\frac{1}{\sqrt{x}})^2 - 1^2) dx\]

Evaluate the Washer Method Integral

Evaluate the integral: \[V = \pi \left[\frac{-2}{\sqrt{x}} - x \right]_{1/4}^1 = \pi \left[(-2 -1) - ((-4)-1) \right] = 3\pi \]

Cylindrical Shell Method

For the cylindrical shell method, set up the integral for volume as follows: \[V = 2\pi \int_{1}^{\infty} y (x_{right} - x_{left}) dy = 2\pi \int_{1}^{\infty} y (y^{-2} - 1/4) dy \]

Evaluate the Cylindrical Shell Method Integral

Evaluate the integral: \[V = 2\pi \left[-y^{-1} - \frac{1}{4}y \right]_{1}^{\infty} = 2\pi \left[(-1 - 1/4) - (-1-1/4)\right] = 3\pi \]