Find the area of the surface generated when the given curve is revolved about the given axis.


y=√1−x^2, for −1/2≤x≤1/2; about the x-axis

Verified step by step guidance

1. Step 1: Recall the formula for the surface area of a curve revolved about the x-axis. The formula is: $A\; =\; 2\pi \; \int [a,b]\; y\; \surd (1\; +\; (dy/dx)^2)\; dx$, where y is the function being revolved, and dy/dx is its derivative.
2. Step 2: Identify the given function and its interval. Here, y = √(1 − x^2) and the interval is −1/2 ≤ x ≤ 1/2.
3. Step 3: Compute the derivative of y with respect to x. Using the chain rule, $dy/dx\; =\; d/dx(\surd (1\; -\; x^2))\; =\; -x\; /\; \surd (1\; -\; x^2)$.
4. Step 4: Substitute y and dy/dx into the surface area formula. Replace y with √(1 − x^2) and dy/dx with −x / √(1 − x^2) in the formula: $A\; =\; 2\pi \; \int [-1/2,1/2]\; \surd (1\; -\; x^2)\; \surd (1\; +\; (-x\; /\; \surd (1\; -\; x^2))^2)\; dx$.
5. Step 5: Simplify the integrand. Combine terms under the square root and simplify: $\surd (1\; +\; (-x\; /\; \surd (1\; -\; x^2))^2)\; =\; \surd (1\; +\; x^2\; /\; (1\; -\; x^2))\; =\; \surd (1\; /\; (1\; -\; x^2))$. The integrand becomes $\surd (1\; -\; x^2)\; *\; \surd (1\; /\; (1\; -\; x^2))\; =\; 1$. The integral simplifies to $A\; =\; 2\pi \; \int [-1/2,1/2]\; 1\; dx$.

Key Concepts

Surface Area of Revolution

The surface area of revolution is calculated by rotating a curve around an axis. The formula for the surface area generated by revolving a function y = f(x) about the x-axis is given by S = 2π ∫[a to b] f(x) √(1 + (f'(x))^2) dx, where f'(x) is the derivative of f(x). This concept is essential for determining the area of the surface created by the rotation of the curve.

Parametric Representation

In some cases, curves can be represented parametrically, which involves expressing the coordinates of points on the curve as functions of a parameter. For the given curve y = √(1 - x^2), it can be helpful to consider its parametric form, where x = cos(t) and y = sin(t) for t in the appropriate range. This representation can simplify the integration process when calculating the surface area.

Definite Integrals

Definite integrals are used to calculate the area under a curve between two specified limits. In the context of finding the surface area of revolution, the definite integral evaluates the accumulated area as the curve is revolved around the axis. Understanding how to set up and compute definite integrals is crucial for solving problems related to surface areas.