let C be the curve y= 7sqrtx for 1.2 < x <3.2 find the surface area of revolution about the x-axis of R 


To find the surface area of revolution of the curve \( y = 7\sqrt{x} \) about the x-axis, we can use the formula for the surface area of revolution:

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

### Step-by-step solution:

1. **Given function:**
   \[
   y = 7\sqrt{x}
   \]
   We need to compute \( \frac{dy}{dx} \) (the derivative of \( y \) with respect to \( x \)).

2. **Derivative \( \frac{dy}{dx} \):**
   \[
   y = 7x^{1/2}
   \]
   Using the power rule for differentiation:
   \[
   \frac{dy}{dx} = 7 \times \frac{1}{2}x^{-1/2} = \frac{7}{2\sqrt{x}}
   \]

3. **Surface Area formula:**
   Now, substitute \( y = 7\sqrt{x} \) and \( \frac{dy}{dx} = \frac{7}{2\sqrt{x}} \) into the surface area formula:
   \[
   S = 2\pi \int_{1.2}^{3.2} 7\sqrt{x} \sqrt{1 + \left( \frac{7}{2\sqrt{x}} \right)^2} \, dx
   \]
   
4. **Simplifying inside the square root:**
   \[
   1 + \left( \frac{7}{2\sqrt{x}} \right)^2 = 1 + \frac{49}{4x}
   \]
   So, the integral becomes:
   \[
   S = 2\pi \int_{1.2}^{3.2} 7\sqrt{x} \sqrt{1 + \frac{49}{4x}} \, dx
   \]

This integral is a bit complex and is best evaluated using numerical methods.

I can compute this definite integral to find the surface area numerically. Let's proceed with that.The surface area of revolution of the curve \( y = 7\sqrt{x} \) about the x-axis, for \( 1.2 < x < 3.2 \), is approximately **334.31 square units**.

Would you like further details on any step or have any questions?

Here are some related questions for practice:

1. What is the surface area of revolution for the curve \( y = x^2 \) about the x-axis between \( x = 1 \) and \( x = 4 \)?
2. How would the surface area change if the curve is rotated about the y-axis instead?
3. Can you compute the surface area of revolution for \( y = \frac{1}{x} \) between \( x = 2 \) and \( x = 5 \)?
4. What is the arc length of \( y = 7\sqrt{x} \) between \( x = 1.2 \) and \( x = 3.2 \)?
5. What happens to the surface area if the curve is \( y = 7\sqrt{x+1} \)?

**Tip:** When integrating functions with square roots, double-check the algebraic manipulations under the square root for accuracy.

Let C be the curve y= 7sqrtx for 1.2 < x <3.2. Find the surface area of revolution about the x-axis of R.

Learn how to calculate the surface area of revolution for the curve y = 7√x about the x-axis between x = 1.2 and x = 3.2. This problem involves using calculus concepts such as integration and differentiation, and is suitable for students in Grades 11-12 or introductory college calculus.

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