Find the exact area of the surface obtained by rotating the curve about the *x*-axis.

*x* =

1

3

(*y*2 + 2)3/2,    3 ≤ *y* ≤ 4

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!

Find the exact area of the surface obtained by rotating the curve about the x-axis.

x = (1/3)(y^2 + 2)^(3/2), 3 ≤ y ≤ 4

Explore how to calculate the surface area of revolution for the curve x = (1/3)(y^2 + 2)^(3/2) rotated about the x-axis. This detailed solution involves calculus concepts, including differentiation, integration, and applying the surface area formula. Suitable for college-level students studying calculus.

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