Find the exact area of the surface obtained by rotating the curve about the x-axis:   x = 2 + 3y^2 , 1 ≤ y ≤ 2


To find the exact area of the surface obtained by rotating the curve \( x = 2 + 3y^2 \), where \( 1 \leq y \leq 2 \), about the \( x \)-axis, we use the formula for the surface area of a curve rotated about the \( x \)-axis. In terms of \( y \), the surface area \( A \) is given by:

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

where:
- \( r(y) = x(y) = 2 + 3y^2 \) is the radius (distance from the x-axis to the curve at a given point),
- \( \frac{dx}{dy} \) is the derivative of \( x \) with respect to \( y \),
- \( [y_1, y_2] = [1, 2] \).

### Step 1: Compute \( \frac{dx}{dy} \)
\[
x = 2 + 3y^2
\]
Differentiating \( x \) with respect to \( y \), we get:
\[
\frac{dx}{dy} = 6y
\]

### Step 2: Substitute into the surface area formula
Now we substitute into the surface area formula:

\[
A = 2\pi \int_1^2 (2 + 3y^2) \sqrt{1 + (6y)^2} \, dy
\]

Simplifying inside the square root:

\[
\sqrt{1 + (6y)^2} = \sqrt{1 + 36y^2}
\]

So the integral becomes:

\[
A = 2\pi \int_1^2 (2 + 3y^2) \sqrt{1 + 36y^2} \, dy
\]

### Step 3: Compute the integral
At this point, this integral is not elementary, and it requires special techniques, such as substitution or numerical methods, to evaluate. However, in an exact form, the surface area is expressed as:

\[
A = 2\pi \int_1^2 (2 + 3y^2) \sqrt{1 + 36y^2} \, dy
\]

This is the exact expression for the surface area. To find the numerical result, one could use a calculator or software to compute the value of this integral.

Would you like a more detailed step-by-step approach or numerical evaluation?

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Here are 5 related questions that could expand your understanding:
1. How do you evaluate integrals with square roots involving polynomials?
2. What is the general formula for the surface area of a curve rotated about the y-axis?
3. How does changing the axis of rotation affect the surface area formula?
4. What role does the derivative play in surface area problems involving rotation?
5. Can the surface area formula be applied to parametric curves?

**Tip:** When dealing with complex integrals, sometimes numerical methods (such as using a calculator or software like WolframAlpha) are the best approach for obtaining exact or approximate values.

Find the exact area of the surface obtained by rotating the curve about the x-axis: x = 2 + 3y^2 , 1 ≤ y ≤ 2

Discover how to compute the exact surface area of revolution for the curve x = 2 + 3y^2 when rotated about the x-axis over the interval 1 ≤ y ≤ 2. This solution involves differentiation, definite integrals, and the surface area formula for a curve rotated about an axis. Suitable for college-level calculus students.

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