1.5: Surface Area of Revolution

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Oct 27, 2024
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- 1.4: Arc Length
- 1.6: The Volume of Cored Sphere

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The area of a frustum is

$A = 2\pi r(length). \nonumber$

If we revolve a curve around the x-axis, we have that the surface area of revolution is given by

$\text{Area} = 2\pi \int _a^b y \sqrt{1+\left( \dfrac{dy}{dx} \right)^2} dx. \nonumber$

Example 1

Set up an integral that gives the surface area of revolution about the x axis of the curve

$y = x^2 \nonumber$

from 2 to 3.

Solution

We find

$\left(\dfrac{dy}{dx} \right)^2=(2x)^2 = 4x^2. \nonumber$

Now use the area formula:

$A = 2\pi\int_2^3 x^2\sqrt{1+4x^2} dx. \nonumber$

We will learn later how to work out this integral. However a computer gives that

$A \approx 208.09. \nonumber$

Larry Green (Lake Tahoe Community College)

Integrated by Justin Marshall.

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