Understanding the Disk Method for Calculating the Volume of Solids of Revolution

When dealing with the volume of solids obtained by rotating regions bounded by given curves about a specified axis, the disk method is a powerful tool in calculus. This method allows us to break down complex three-dimensional shapes into simpler geometric solids, such as disks or washers, for easier integration.

Defining the Region of Interest

To begin with, let's define the region we are interested in. In this case, we are given the curves y sqrt{x - 1}, y 0, and the vertical line x 4, and we are asked to find the volume of the solid formed by rotating this region about the x-axis.

Step 1: Identify the Region of Integration

The region is defined by the following:- The curve y sqrt{x - 1}- The x-axis (where y 0)- The vertical line x 4To find where the curve intersects the x-axis, we solve the equation:y 0sqrt{x - 1} 0Substituting 0 into y:x - 1 0 rarr; x 1Thus, the region of integration is between x 1 and x 4.

Step 2: Set Up the Volume Integral

The volume V can be calculated using the disk method. The volume element at each point in the region is a disk with radius y and thickness dx. Therefore, the volume is given by the integral:[ V pi int_{a}^{b} [f(x)]^2 dx ]Here, the radius of each disk is y sqrt{x - 1}, and the integration limits are from a 1 to b 4. Substituting the radius into the integral, we get:[ V pi int_{1}^{4} (sqrt{x - 1})^2 dx ]Simplifying the integrand:[ V pi int_{1}^{4} (x - 1) dx ]

Step 3: Evaluate the Integral

To evaluate the integral, we compute:[ V pi left[ frac{x^2}{2} - x right]_{1}^{4} ]Now, substituting the limits of integration:- At x 4[ frac{4^2}{2} - 4 8 - 4 4 ]- At x 1[ frac{1^2}{2} - 1 frac{1}{2} - 1 -frac{1}{2} ]Thus, the volume is:[ V pi left[ 4 - left( -frac{1}{2} right) right] pi left[ 4 frac{1}{2} right] pi left[ frac{8}{2} frac{1}{2} right] pi left[ frac{9}{2} right] ]Therefore, the volume of the solid obtained by rotating the region about the x-axis is:[ V frac{9pi}{2} ]

Validation and Considerations

It's essential to verify the given equations and constraints to ensure accuracy. If y sqrt{x - 1} is indeed the correct curve, then at x 0, y is imaginary, which is not a valid scenario for physical applications. Thus, the correct region must be from x 1 to x 4 as we have already verification, we can use the standard formula for volumes of revolution and confirm our result:[ V pi int_{1}^{4} (sqrt{x - 1})^2 dx pi int_{1}^{4} (x - 1) dx ]Evaluating this integral again:[ V pi left[ frac{x^2}{2} - x right]_{1}^{4} pi left[ 4 - left( -frac{1}{2} right) right] pi left[ 4 frac{1}{2} right] pi left[ frac{9}{2} right] ]This confirms that the volume is indeed:[ V frac{9pi}{2} ]