Dr. Neustadter
2.4 Arc Length and Surface Area
Derivation of the Arc Length Formula
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Math 1338
Dr. Neustadter
2.4 Arc Length and Surface Area
Summary of Formulas
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Math 1338
Dr. Neustadter
2.4 Arc Length and Surface Area
Math 1338
Example 1
Find the length of the arc of the semicubical parabola 𝑦 2 = 𝑥 3 between the points (1,1) and (4,8).
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Dr. Neustadter
2.4 Arc Length and Surface Area
Example 2.18 in Textbook
Let 𝑓(𝑥) = 2𝑥 3/2. Calculate the arc length of the function 𝑓(𝑥) over the interval [0,1].
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Math 1338
Dr. Neustadter
2.4 Arc Length and Surface Area
Area of Surface of Revolution
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Math 1338
Dr. Neustadter
2.4 Arc Length and Surface Area
Math 1338
Example 1
The curve 𝑦 = √4 − 𝑥 2 , −1 ≤ 𝑥 ≤ 1, is an arc of the circle 𝑥 2 + 𝑦 2 = 4. Find the area of the surface
obtained by rotating this arc about the x-axis.
To visualize the surface area, go to https://www.geogebra.org/m/tWneyTxW.
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Dr. Neustadter
2.4 Arc Length and Surface Area
Math 1338
Example 2
The arc of the parabola 𝑦 = 𝑥 2 from (1,1) to (2,4) is rotated about the y-axis. Find the area of the
resulting surface.
To visualize the surface area, go to https://www.geogebra.org/m/tWneyTxW.
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Dr. Neustadter
2.4 Arc Length and Surface Area
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Math 1338