To determine the volume of the solid of revolution formed by rotating a curve about an axis, we commonly use methods like the Disk Method or the Washer Method. In this specific problem, we are given a simple curve, y = 2, which is a horizontal line. This line is rotated 360° about the x-axis between the limits x = 0 and x = 3.

Volume Calculation using the Disk Method

When a region is rotated about the x-axis, and the solid generated has no hole, the Disk Method is the most suitable approach. The formula for the volume V using the Disk Method for a function y = f(x) rotated about the x-axis from x = a to x = b is given by:

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

Let's break down the components given in the problem:

• The function is \(f(x) = y = 2\).
• The axis of revolution is the x-axis.
• The lower limit of integration is \(a = 0\).
• The upper limit of integration is \(b = 3\).

Applying the Disk Method to the Curve y=2

Now, we will substitute these values into the Disk Method formula to calculate the volume of the solid of revolution.

Step 1: Set up the integral.

Substitute \(f(x) = 2\), \(a = 0\), and \(b = 3\) into the volume formula:

\[V = \int_{0}^{3} \pi (2)^2 dx\]

Step 2: Simplify the integrand.

Square the function value:

\[V = \int_{0}^{3} \pi (4) dx\]

Rearrange the constant:

\[V = 4\pi \int_{0}^{3} dx\]

Step 3: Evaluate the integral.

The integral of \(dx\) is \(x\). So, we evaluate \(x\) from \(0\) to \(3\):

\[V = 4\pi [x]_{0}^{3}\]

Step 4: Apply the limits of integration.

Substitute the upper limit and subtract the substitution of the lower limit:

\[V = 4\pi (3 - 0)\]

Step 5: Calculate the final volume.

\[V = 4\pi (3)\]

\[V = 12\pi\]

The volume of the solid of revolution formed is \(12\pi\) cubic units. This solid is essentially a cylinder with radius 2 and height 3. The volume of a cylinder is given by \(V = \pi r^2 h\). Here, \(r=2\) and \(h=3\), so \(V = \pi (2^2)(3) = \pi (4)(3) = 12\pi\), which confirms our integration result.