[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 = x²/4, 0 ≤ x ≤ 2

Verified step by step guidance

1. Identify the formula for the surface area of a solid of revolution about the x-axis: \[ S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
2. Given the curve \[ y = \frac{x^2}{4} \], find its derivative with respect to \[ x \]: \[ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{x^2}{4} \right) = \frac{x}{2} \]
3. Substitute \[ y \] and \[ \frac{dy}{dx} \] into the surface area formula: \[ S = \int_0^2 2\pi \left( \frac{x^2}{4} \right) \sqrt{1 + \left( \frac{x}{2} \right)^2} \, dx \]
4. Simplify the expression inside the square root and the integrand: \[ S = \int_0^2 2\pi \frac{x^2}{4} \sqrt{1 + \frac{x^2}{4}} \, dx = \int_0^2 \frac{\pi x^2}{2} \sqrt{1 + \frac{x^2}{4}} \, dx \]
5. Evaluate the integral using an appropriate method (such as substitution or numerical integration) to find the surface area, then round the result 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 involves integrating 2π times the radius (the y-value) times the arc length element. This concept connects geometry with calculus to measure curved surfaces.

Arc Length Element (ds)

The arc length element ds represents a small segment of the curve and is given by √(1 + (dy/dx)²) dx. It accounts for the curve's slope, ensuring the surface area calculation accurately follows the curve's shape rather than just the x-interval.

Definite Integration with Limits

Definite integration calculates the exact accumulated value over an interval, here from x = 0 to x = 2. Applying limits ensures the surface area corresponds precisely to the specified portion of the curve, providing a numerical result to two decimal places.