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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.1.77
Chapter 8, Problem 8.1.77

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

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1
Step 1: Recall the formula for the surface area of a solid of revolution about the x-axis: S = 2π ∫[a,b] f(x) √(1 + (f'(x))²) dx. Here, f(x) = √(x + 1), and the interval is [0, 1].
Step 2: Compute the derivative of f(x). Using the chain rule, f'(x) = (1/2)(x + 1)-1/2. Simplify this to f'(x) = 1 / (2√(x + 1)).
Step 3: Substitute f(x) and f'(x) into the surface area formula. The integrand becomes 2π √(x + 1) √(1 + (1 / (2√(x + 1)))²). Simplify the expression inside the square root.
Step 4: Simplify the term 1 + (1 / (2√(x + 1)))². This becomes 1 + 1 / (4(x + 1)). Combine terms under a common denominator if necessary.
Step 5: Set up the integral S = 2π ∫[0,1] √(x + 1) √(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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Example 1: Minimizing Surface Area

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.
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Definition of the Definite Integral

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.
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