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Calculus: Arc Length & Surfaces

This document summarizes the key concepts about computing arc length and surfaces of revolution from multi-variable calculus. It discusses approximating arc length as the sum of line segments, and defines arc length precisely using a definite integral. It also explains how to compute the area of a surface of revolution by treating it as the lateral surface area of a right circular cone, and derives a formula to calculate the surface area as a definite integral involving the revolved curve's radius and arc length. An example calculates the surface area of a curve revolved around the x-axis.

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Lynn Hollenbeck Breindel
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117 views7 pages

Calculus: Arc Length & Surfaces

This document summarizes the key concepts about computing arc length and surfaces of revolution from multi-variable calculus. It discusses approximating arc length as the sum of line segments, and defines arc length precisely using a definite integral. It also explains how to compute the area of a surface of revolution by treating it as the lateral surface area of a right circular cone, and derives a formula to calculate the surface area as a definite integral involving the revolved curve's radius and arc length. An example calculates the surface area of a curve revolved around the x-axis.

Uploaded by

Lynn Hollenbeck Breindel
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Computing Arc Length and Surfaces of Revolution

Multi-Variable Calculus & Calculus Review Chapter 7 Section 4


Mrs. Shak

7.4: Arc Length & Surfaces


Using definite integrals to compute arc length of curves Line segments are given by the familiar Distance Formula

Rectifiable curve = curve with finite arc length Function continuously differentiable on [a,b], i.e. f continuous on [a,b] Graph of f on [a,b] is a Smooth Curve

7.4 : Approximating Arc Length


Arc Length (s) = Sum of little line segments

7.4 : Definition of Arc Length


Computing Arc Length as Definite Integral

7.4: Homework Problem p.483 #4


Find the arc length of the graph of the function over the 3 interval [0,9] 2

y = 2x

+3

Differentiate: Definite Integral: Integrate:

7.4 Area of a Surface of Revolution

The area of a surface of revolution is derived from the formula for the lateral surface area of the frustum of a right circular cone.

NOTE: frustrum is the conic shape (normally cone or pyramid) with the top chopped off by a plane parallel to the base

7.4 Area of Surface of Revolution

How to set up lateral surface area of cylinder as an integral? Can we extend this to the surface area of an object generated by revolving an arbitrary continuous function revolved about the x-axis? What is our new dx, or rather ds?

7.4 Area of a Surface of Revolution


Suppose the graph of a function f, having a continuous derivative on the interval [a, b], is revolved about the xaxis to form a surface of revolution, as shown.

L = x 2 + y 2 y = 1 + x x dy 1 + dx = 1 + f '( x )2 dx = ds dx
2 2

Arclength Formula!!

7.4 Area of a Surface of Revolution

7.4 Area of a Surface of Revolution


In these two formulas for S, you can regard the products 2f(x) and 2x as the circumferences of the circles traced by a point (x, y) on the graph of f as it is revolved about the x-axis and the y-axis (Figure 7.45). In one case the radius is r = f(x), and in the other case the radius is r = x.

Figure 7.45

Example The Area of a Surface of Revolution


Find the area of the surface formed by revolving the graph of f(x) = x3 on the interval [0, 1] about the x-axis, as shown.

Figure 7.46

Example Solution
The distance between the x-axis and the graph of f is r(x) = f(x), and because f'(x) = 3x2, the surface area is

Homework
Read Chapter 7.4 (Single Var Textbook) Do Homework
Pg 484 (39-46:all, 51, 53): 10 problems State the function, its derivative, and the complete integral either integrate by hand or on the calculator

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