We are tasked with finding the exact area of the surface generated by rotating the given curve about the x-axis. The curve is described by the function

$x = \frac{1}{3}(y^2 + 2)^{\frac{3}{2}}, \quad 3 \leq y \leq 4.$

To find the surface area of a curve rotated about the x-axis, we use the surface area formula:

$A = 2\pi \int_{a}^{b} y \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy.$

Step 1: Differentiate x with respect to y

We begin by finding $\frac{dx}{dy}$. The given equation is:

$x = \frac{1}{3}(y^2 + 2)^{\frac{3}{2}}.$

Differentiate $x$ with respect to $y$:

$\frac{dx}{dy} = \frac{1}{3} \cdot \frac{d}{dy}\left( (y^2 + 2)^{\frac{3}{2}} \right).$

Using the chain rule:

$\frac{dx}{dy} = \frac{1}{3} \cdot \frac{3}{2} (y^2 + 2)^{\frac{1}{2}} \cdot 2y = y (y^2 + 2)^{\frac{1}{2}}.$

Step 2: Set up the surface area integral

Now, substitute this result into the surface area formula. The surface area is:

$A = 2\pi \int_{3}^{4} y \sqrt{1 + \left( y (y^2 + 2)^{\frac{1}{2}} \right)^2} \, dy.$

Simplify the expression inside the square root:

$A = 2\pi \int_{3}^{4} y \sqrt{1 + y^2 (y^2 + 2)} \, dy.$

Simplify further:

$A = 2\pi \int_{3}^{4} y \sqrt{1 + y^2(y^2 + 2)} = 2\pi \int_{3}^{4} y \sqrt{1 + y^4 + 2y^2} \, dy.$

Thus, the expression inside the square root becomes:

$A = 2\pi \int_{3}^{4} y \sqrt{y^4 + 2y^2 + 1} \, dy = 2\pi \int_{3}^{4} y \sqrt{(y^2 + 1)^2} \, dy.$

Since $\sqrt{(y^2 + 1)^2} = y^2 + 1$, the integral simplifies to:

$A = 2\pi \int_{3}^{4} y (y^2 + 1) \, dy.$

Step 3: Compute the integral

Now, expand the integrand:

$A = 2\pi \int_{3}^{4} (y^3 + y) \, dy.$

Integrate term by term:

$\int (y^3 + y) \, dy = \frac{y^4}{4} + \frac{y^2}{2}.$

Evaluate this from $y = 3$ to $y = 4$:

$A = 2\pi \left[ \left( \frac{4^4}{4} + \frac{4^2}{2} \right) - \left( \frac{3^4}{4} + \frac{3^2}{2} \right) \right].$

Calculate each term:

For $y = 4$:

$\frac{4^4}{4} + \frac{4^2}{2} = \frac{256}{4} + \frac{16}{2} = 64 + 8 = 72.$

For $y = 3$:

$\frac{3^4}{4} + \frac{3^2}{2} = \frac{81}{4} + \frac{9}{2} = 20.25 + 4.5 = 24.75.$

Now subtract:

$72 - 24.75 = 47.25.$

Step 4: Final surface area

Multiply by $2\pi$:

$A = 2\pi \times 47.25 = 94.5\pi.$

Thus, the exact surface area is:

$A = 94.5\pi \text{ square units}.$

Would you like more details on any of the steps, or further clarification?

Here are five related questions you could explore:

1. How does the surface area formula change for rotation about the y-axis?
2. What is the geometric interpretation of $\frac{dx}{dy}$?
3. How would the surface area change if the curve were rotated about the y-axis instead of the x-axis?
4. Can you find the arc length of the same curve for the same range of y?
5. What happens to the surface area if we change the range of y to $[2, 4]$?

Tip: Always simplify the expression inside the square root before integrating; it makes the process much easier!