Find the exact area of the surface obtained by rotating the
curve about the x-axis. y= cos (x/2), 0 less than or equal to x less than or equal to pi

To find the exact area of the surface obtained by rotating the curve \( y = \cos\left(\frac{x}{2}\right) \) about the x-axis over the interval \( [0, \pi] \), we can use the formula for the surface area of a surface of revolution about the x-axis:

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

### Step 1: Compute the derivative \( \frac{dy}{dx} \)

Given \( y = \cos\left(\frac{x}{2}\right) \), we compute its derivative with respect to \( x \):

\[
\frac{dy}{dx} = -\frac{1}{2} \sin\left(\frac{x}{2}\right)
\]

### Step 2: Plug into the surface area formula

Now substitute \( y = \cos\left(\frac{x}{2}\right) \) and \( \frac{dy}{dx} = -\frac{1}{2} \sin\left(\frac{x}{2}\right) \) into the formula for the surface area:

\[
A = 2\pi \int_{0}^{\pi} \cos\left(\frac{x}{2}\right) \sqrt{1 + \left( -\frac{1}{2} \sin\left(\frac{x}{2}\right) \right)^2} \, dx
\]

Simplifying the square root term:

\[
1 + \left( -\frac{1}{2} \sin\left(\frac{x}{2}\right) \right)^2 = 1 + \frac{1}{4} \sin^2\left(\frac{x}{2}\right)
\]

Thus, the integral becomes:

\[
A = 2\pi \int_{0}^{\pi} \cos\left(\frac{x}{2}\right) \sqrt{1 + \frac{1}{4} \sin^2\left(\frac{x}{2}\right)} \, dx
\]

### Step 3: Solve the integral

This is a more challenging integral that typically requires numerical methods or advanced techniques to solve explicitly. However, the general solution for the exact surface area is given by this integral expression. For practical purposes, this integral can be evaluated numerically or left in its exact form.

Would you like me to evaluate this integral numerically, or leave it in the exact form?

---

Here are 5 related questions:

1. How would the surface area formula change if we rotated around the y-axis instead?
2. What happens to the surface area if the function were shifted vertically?
3. How would you calculate the volume of the solid formed by rotating the same curve about the x-axis?
4. What is the interpretation of the surface area in terms of real-world applications?
5. How does the presence of the derivative \( \frac{dy}{dx} \) affect the surface area formula?

**Tip**: When dealing with integrals involving squares and square roots, try simplifying trigonometric identities where possible to reduce complexity.

Find the exact area of the surface obtained by rotating the curve about the x-axis. y = cos(x/2), 0 ≤ x ≤ π

Explore how to calculate the surface area of a revolution generated by rotating the curve y = cos(x/2) around the x-axis over the interval [0, π]. This solution uses calculus techniques such as differentiation and definite integration, ideal for advanced high school or college students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation