Volume of Revolution Explained

When a two-dimensional region is revolved around an axis, it forms a three-dimensional solid. The volume of such a solid, known as a solid of revolution, can be calculated using integration. For revolution around the x-axis, the most common method is the Disk Method.

The parabolic arc given is \(y = \sqrt{x}\), and it is revolved around the x-axis over the interval \(1 \le x \le 2\).

Formula for Volume of Revolution

The volume \(V\) of a solid generated by revolving the region under the curve \(y = f(x)\) from \(x=a\) to \(x=b\) around the x-axis is given by the formula:

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

Applying the Volume Formula to the Parabolic Arc

In this specific problem:

• The function is \(f(x) = \sqrt{x}\).
• The square of the function is \([f(x)]^2 = (\sqrt{x})^2 = x\).
• The lower limit of integration (a) is \(1\).
• The upper limit of integration (b) is \(2\).

Substitute these values into the volume formula:

$$V = \pi \int_{1}^{2} x \, dx$$

Calculating the Definite Integral

Now, we need to evaluate the definite integral.

1. Integrate \(x\): The integral of \(x\) with respect to \(x\) is \(\frac{x^2}{2}\).
2. Apply the Limits of Integration: Evaluate the antiderivative at the upper limit (2) and subtract its value at the lower limit (1).

$$V = \pi \left[ \frac{x^2}{2} \right]_{1}^{2}$$

$$V = \pi \left( \frac{(2)^2}{2} - \frac{(1)^2}{2} \right)$$

$$V = \pi \left( \frac{4}{2} - \frac{1}{2} \right)$$

$$V = \pi \left( 2 - \frac{1}{2} \right)$$

To subtract the fractions, find a common denominator:

$$V = \pi \left( \frac{4}{2} - \frac{1}{2} \right)$$

$$V = \pi \left( \frac{4 - 1}{2} \right)$$

$$V = \pi \left( \frac{3}{2} \right)$$

$$V = \frac{3\pi}{2}$$

Final Volume of the Solid

The volume of the solid of revolution generated by revolving the parabolic arc \(y = \sqrt{x}\) from \(x = 1\) to \(x = 2\) around the x-axis is \(\frac{3\pi}{2}\) cubic units.