Calculating the Volume of Revolution about the X-Axis

The problem asks us to find the volume generated when the curve defined by the equation \(y = \sqrt {1 + {x^2}} \) is revolved around the x-axis. The arc lies between the limits x = 0 and x = 4. This is a classic problem involving the calculation of the Volume of Revolution using integral calculus.

Formula for Volume of Revolution

When a curve \(y = f(x)\) from x = a to x = b is revolved about the x-axis, the volume of the solid generated is given by the formula:

\(V = \pi \int_{a}^{b} [f(x)]^2 dx\)

In our case, the function is \(y = \sqrt {1 + {x^2}}\), and the limits are a = 0 and b = 4. So, \(y^2 = (\sqrt {1 + {x^2}})^2 = 1 + x^2\).

Setting up the Integral

Substituting the function and the limits into the volume formula, we get the integral for the Volume of Revolution:

\(V = \pi \int_{0}^{4} (1 + x^2) dx\)

This definite integral will give us the required volume.

Evaluating the Integral for Volume of Revolution

Now, we need to evaluate the integral. We use the power rule for integration, which is a fundamental concept in calculus.

\(\int (1 + x^2) dx = \int 1 dx + \int x^2 dx\)

\(= x + \frac{x^{2+1}}{2+1} + C\)

\(= x + \frac{x^3}{3} + C\)

Now, we apply the limits of integration from 0 to 4:

\(V = \pi \left[ x + \frac{x^3}{3} \right]_{0}^{4}\)

Substitute the upper limit (x=4) and subtract the result of substituting the lower limit (x=0).

\(V = \pi \left[ \left( 4 + \frac{4^3}{3} \right) - \left( 0 + \frac{0^3}{3} \right) \right]\)

\(V = \pi \left[ \left( 4 + \frac{64}{3} \right) - \left( 0 + 0 \right) \right]\)

\(V = \pi \left[ 4 + \frac{64}{3} \right]\)

To add the terms inside the bracket, find a common denominator:

\(4 = \frac{4 \times 3}{3} = \frac{12}{3}\)

\(V = \pi \left[ \frac{12}{3} + \frac{64}{3} \right]\)

\(V = \pi \left[ \frac{12 + 64}{3} \right]\)

\(V = \pi \left[ \frac{76}{3} \right]\)

\(V = \frac{76\pi}{3}\)

The resulting volume generated by revolving the arc about the x-axis is \(\frac{76\pi}{3}\). This calculation demonstrates the application of integral calculus to find the Volume of Revolution. Understanding how to set up and evaluate the integral is crucial for such problems involving the x-axis.

Let's summarize the steps:

1. Identify the function \(y = f(x)\) and the limits of integration a and b.
2. Square the function to get \(y^2 = [f(x)]^2\).
3. Set up the definite integral using the formula \(V = \pi \int_{a}^{b} y^2 dx\).
4. Evaluate the integral using techniques from integral calculus.
5. Calculate the value at the upper and lower limits and find the difference.

This systematic approach helps in solving problems related to the Volume of Revolution about the x-axis.