Find the exact area of the surface obtained by rotating the curve about the x-axis.
x = 5 + 4y2,    1 ≤ y ≤ 2

To find the exact area of the surface obtained by rotating the curve about the x-axis, we use the formula for the surface area of revolution about the x-axis:

\[
A = 2\pi \int_{y_1}^{y_2} r(y) \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy
\]

Where:
- \( r(y) \) is the radius of rotation, which in this case is \( x(y) \).
- \( \frac{dx}{dy} \) is the derivative of \( x \) with respect to \( y \).

### Step 1: Derive \( \frac{dx}{dy} \)
The given equation is \( x = 5 + 4y^2 \). Let's find the derivative of \( x \) with respect to \( y \):

\[
\frac{dx}{dy} = \frac{d}{dy}(5 + 4y^2) = 8y
\]

### Step 2: Set up the surface area integral
Now, substitute the expressions for \( x(y) \) and \( \frac{dx}{dy} \) into the surface area formula:

\[
A = 2\pi \int_{1}^{2} (5 + 4y^2) \sqrt{1 + (8y)^2} \, dy
\]

\[
A = 2\pi \int_{1}^{2} (5 + 4y^2) \sqrt{1 + 64y^2} \, dy
\]

### Step 3: Solve the integral
To find the exact area, we can either apply numerical methods or substitution to simplify the integral further. Let me calculate this step for you.It looks like there was a timeout while computing the integral. Let's break it down and focus on solving the integral in parts manually. I'll guide you through it to help you understand.

The integral we need to solve is:

\[
A = 2\pi \int_{1}^{2} (5 + 4y^2) \sqrt{1 + 64y^2} \, dy
\]

This integral is complex and would usually be evaluated numerically. Here’s how you would approach it:

1. **Substitution method**: Let’s try a substitution like \( u = 1 + 64y^2 \), which simplifies the square root. Then, we can express \( dy \) in terms of \( du \), which simplifies the integration. 

Alternatively, this could also be solved using a numerical integration method to approximate the area. Would you prefer to go the numerical route, or would you like a step-by-step breakdown on solving this by substitution?

Find the exact area of the surface obtained by rotating the curve about the x-axis.
x = 5 + 4y^2, 1 ≤ y ≤ 2

Learn how to calculate the exact surface area of revolution for the curve x = 5 + 4y^2, rotated about the x-axis from y = 1 to y = 2. This problem involves key calculus concepts such as derivatives, definite integrals, and the surface area of revolution formula. Suitable for advanced high school and early college students.

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