Let f(x) = √(x + 1). Find the area of the surface generated when:
Region bounded by f(x) and the x-axis on [0, 1]
Revolved about the x-axis

Verified step by step guidance

1. Step 1: Recall the formula for the surface area of a solid of revolution about the x-axis: $S\; =\; 2\pi \; \int [a,b]\; f(x)\; \surd (1\; +\; (f\text{'}(x))²)\; dx$. Here, f(x) = √(x + 1), and the interval is [0, 1].
2. Step 2: Compute the derivative of f(x). Using the chain rule, $f\text{'}(x)\; =\; (1/2)(x\; +\; 1)-1/2$. Simplify this to $f\text{'}(x)\; =\; 1\; /\; (2\surd (x\; +\; 1))$.
3. Step 3: Substitute f(x) and f'(x) into the surface area formula. The integrand becomes $2\pi \; \surd (x\; +\; 1)\; \surd (1\; +\; (1\; /\; (2\surd (x\; +\; 1)))²)$. Simplify the expression inside the square root.
4. Step 4: Simplify the term $1\; +\; (1\; /\; (2\surd (x\; +\; 1)))²$. This becomes $1\; +\; 1\; /\; (4(x\; +\; 1))$. Combine terms under a common denominator if necessary.
5. Step 5: Set up the integral $S\; =\; 2\pi \; \int [0,1]\; \surd (x\; +\; 1)\; \surd (1\; +\; 1\; /\; (4(x\; +\; 1)))\; dx$. Evaluate this integral using appropriate techniques, such as substitution or numerical methods, depending on the complexity of the integrand.

Key Concepts

Surface Area of Revolution

The surface area of revolution is calculated by rotating a curve around an axis. For a function f(x) revolved around the x-axis, the formula involves integrating the circumference of infinitesimally thin rings formed by the rotation. The formula is given by A = 2π ∫ 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 generated by the revolution.

Definite Integral

A definite integral calculates the accumulation of quantities, such as area, over a specific interval [a, b]. In this context, it is used to find the area of the surface generated by revolving the function f(x) around the x-axis from x = 0 to x = 1. The definite integral provides a numerical value representing the total area, which is crucial for solving the problem.

Derivative and Its Role

The derivative of a function, denoted as f'(x), represents the rate of change of the function with respect to x. In the context of finding the surface area of revolution, the derivative is used to compute the term √(1 + (f'(x))²), which accounts for the slope of the curve. Understanding how to find and apply derivatives is vital for accurately calculating the surface area.