find the surface area of the sqrt(3x) on the range 0-9

To find the surface area of the curve \( y = \sqrt{3x} \) from \( x = 0 \) to \( x = 9 \), we use the formula for the surface area of a curve when revolved around the x-axis:

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

### Step 1: Find \(\frac{dy}{dx}\)
First, we differentiate \( y = \sqrt{3x} \):

\[
y = (3x)^{1/2}
\]
\[
\frac{dy}{dx} = \frac{1}{2} (3x)^{-1/2} \cdot 3 = \frac{3}{2\sqrt{3x}}
\]

### Step 2: Set up the surface area formula
Now substitute \( y = \sqrt{3x} \) and \( \frac{dy}{dx} = \frac{3}{2\sqrt{3x}} \) into the surface area formula:

\[
A = 2\pi \int_{0}^{9} \sqrt{3x} \cdot \sqrt{1 + \left(\frac{3}{2\sqrt{3x}}\right)^2} \, dx
\]

### Step 3: Simplify the expression inside the square root
Simplify \( \left(\frac{3}{2\sqrt{3x}}\right)^2 \):

\[
\left(\frac{3}{2\sqrt{3x}}\right)^2 = \frac{9}{12x} = \frac{3}{4x}
\]

Thus, the surface area formula becomes:

\[
A = 2\pi \int_{0}^{9} \sqrt{3x} \cdot \sqrt{1 + \frac{3}{4x}} \, dx
\]

### Step 4: Solve the integral
This integral is quite complex and requires some advanced methods, possibly substitution or numerical techniques. Let me compute the result for you.The surface area of the curve \( y = \sqrt{3x} \) when revolved around the x-axis from \( x = 0 \) to \( x = 9 \) is:

\[
A = 2\pi \left( -\frac{3}{4} + \frac{39\sqrt{13}}{4} \right)
\]

This can be approximated numerically as approximately 336.05 square units.

Would you like a detailed step-by-step breakdown or any further clarifications?

Here are 5 related questions to deepen your understanding:
1. How does the surface area formula change when rotating around the y-axis instead?
2. What is the surface area of \( y = \sqrt{2x} \) over the same range?
3. Can we compute surface area for non-functional curves, like circles or ellipses?
4. How does the surface area change if we change the range from 0 to 9 to a different interval?
5. What happens if you revolve the curve around a line other than the x-axis?

**Tip:** For more complex integrals, symbolic methods may simplify the process, but numeric approximations are often used in practice.

Calculate the surface area of the curve y = sqrt(3x) when revolved around the x-axis between x = 0 and x = 9. This problem covers essential calculus concepts, including differentiation, the surface area formula, and definite integrals. The result is approximated to 336.05 square units. Suitable for Grades 11-12 or early college-level calculus students.

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