Determine the volume of the solid of revolution formed when the curve y = 2 is rotated 360° about the x-axis between the limits x = 0 to x = 3.

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.