Chapter 7: Additional Applications of Integration
Section 7.3: The Theorems of Pappus
Example 7.3.1
Use the first theorem of Pappus to find the volume of the solid of revolution formed when the plane region bounded by the x-axis and the graphs of y=x2 and x=1, is rotated about the x-axis. (See Example 5.2.1.)
Solution
Mathematical Solution
The area of the plane region is ∫01x2 ⅆx=13.
Since the rotation is about the y-axis, the radius of rotation for the centroid is y&conjugate0;=11/3∫0112x22 ⅆx=310.
Hence, the centroid traverses a distance of 2 π3/10=3 π/5, so by the first theorem of Pappus, the volume of the solid of revolution is 3 π/5⋅1/3=π/5, as found in Example 5.2.1.
Maple Solution
Initialize
Context Panel: Assign Name
f=x2→assign
Calculate area
Expression palette: Definite Integral template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻A
∫01f ⅆx = 13→assign to a nameA
Determine y&conjugate0;
Apply the formula for the centroid of a plane region. See the lower-left portion of Table 5.7.1.
1A∫01f2/2 ⅆx = 310→assign to a nameYbar
Apply the first theorem of Pappus
Multiply the circumference of the circle whose radius is y&conjugate0; by the area A.
2 π Ybar⋅A = 15π
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