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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.6.13
Chapter 6, Problem 6.6.13

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

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

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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. 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.
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Example 1: Minimizing Surface Area

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.
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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.
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Related Practice
Textbook Question

Determine the area of the shaded region in the following figures.

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Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = √x,y=0, and x=4; about the x-axis

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Textbook Question

A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume x and y are measured in meters. 


The spherical zone generated when the curve y=√8x−x^2 on the interval 1≤x≤7 is revolved about the x-axis 

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Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.


y=|x| and y=2−x^2; about the x-axis 

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Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 


y = x³−x⁸+1,y=1; about the y-axis

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Textbook Question

Determine the area of the shaded region in the following figures.

(Hint: Find the intersection point by inspection.)

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