[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = sin x, 0 ≤ x ≤ π

Verified step by step guidance

1. Identify the formula for the surface area of a curve revolved around the x-axis. The surface area \( S \) is given by: \[ S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
2. For the given curve \( y = \sin x \), determine the interval of integration, which is from \( a = 0 \) to \( b = \pi \).
3. Compute the derivative of \( y \) with respect to \( x \): \[ \frac{dy}{dx} = \cos x \]
4. Substitute \( y = \sin x \) and \( \frac{dy}{dx} = \cos x \) into the surface area formula: \[ S = \int_0^{\pi} 2\pi \sin x \sqrt{1 + \cos^2 x} \, dx \]
5. Set up the integral for evaluation, and then use a calculator or computer to approximate the value of the integral to two decimal places.

Key Concepts

Surface Area of Revolution

The surface area of a solid formed by revolving a curve around an axis is found using an integral formula. For revolution about the x-axis, the formula is S = ∫ 2πy √(1 + (dy/dx)²) dx over the given interval. This calculates the total area of the curved surface generated.

Derivative of the Function

To apply the surface area formula, you need the derivative dy/dx of the function y = sin x. The derivative, cos x, measures the slope of the curve at each point and is essential for computing the integrand's square root term, which accounts for the curve's steepness.

Definite Integration and Numerical Approximation

Evaluating the surface area requires integrating the function from 0 to π. Since the integral may not have a simple closed form, numerical methods or a calculator are used to approximate the value to two decimal places, ensuring an accurate and practical solution.