Questions: Use the Shell Method to compute the volume of the solid obtained by rotating the region underneath the graph of (y=frac1sqrtx^2+10) over the interval ([0,5]), about (x=0).

$Use the Shell Method to compute the volume of the solid obtained by rotating the region underneath the graph of (y=frac1sqrtx^2+10) over the interval ([0,5]), about (x=0).$

Transcript text: Use the Shell Method to compute the volume of the solid obtained by rotating the region underneath the graph of $y=\frac{1}{\sqrt{x^{2}+10}}$ over the intenal $[0,5]$, about $x=0$.

Solution

Solution Steps

To solve this problem using the Shell Method, we need to set up the integral for the volume of the solid of revolution. The Shell Method involves integrating the product of the circumference of the shell, the height of the shell, and the thickness of the shell. Here, the height of the shell is given by the function $ y = \frac{1}{\sqrt{x^2 + 10}} $, the radius is $ x $, and the thickness is $ dx $. The integral will be evaluated from 0 to 5.

Solution Approach

Identify the radius of the shell, which is $ x $.
Identify the height of the shell, which is $ \frac{1}{\sqrt{x^2 + 10}} $.
Set up the integral for the volume using the Shell Method formula: $ V = 2\pi \int_{a}^{b} (radius \times height) \, dx $.
Evaluate the integral from 0 to 5.

Step 1: Identify the Radius and Height of the Shell

The radius of the shell is $ x $ and the height of the shell is given by the function $ y = \frac{1}{\sqrt{x^2 + 10}} $.

Step 2: Set Up the Integral

Using the Shell Method, the volume $ V $ of the solid of revolution is given by: \[ V = 2\pi \int_{0}^{5} x \cdot \frac{1}{\sqrt{x^2 + 10}} \, dx \]

Step 3: Evaluate the Integral

The integral can be evaluated as follows: \[ V = 2\pi \left[ -\sqrt{10} + \sqrt{35} \right] \]

Step 4: Simplify the Expression

Simplifying the expression, we get: \[ V = 2\pi \left( \sqrt{35} - \sqrt{10} \right) \]

Step 5: Calculate the Numerical Value

Evaluating the numerical value, we find: \[ V \approx 17.3026 \]