Chapter 6 | Applications of Integration
717
kinds of solids by using the centroid. (There is also a theorem of Pappus for surface area, but it is much less useful than the theorem for volume.)
Theorem 6.14: Theorem of Pappus for Volume Let R be a region in the plane and let l be a line in the plane that does not intersect R . Then the volume of the solid of revolution formed by revolving R around l is equal to the area of R multiplied by the distance d traveled by the centroid of R.
Proof We can prove the case when the region is bounded above by the graph of a function f ( x ) and below by the graph of a function g ( x ) over an interval ⎡ ⎣ a , b ⎤ ⎦ , and for which the axis of revolution is the y -axis. In this case, the area of the region is A = ∫ ⎡ ⎣ f ( x )− g ( x ) ⎤ ⎦ dx . Since the axis of rotation is the y -axis, the distance traveled by the centroid of the region depends
a b
only on the x -coordinate of the centroid, x – , which is
M y
x – =
m ,
where
a b
⎦ dx and M y = ρ ∫ a b x ⎡
m = ρ ∫
⎡ ⎣ f ( x )− g ( x ) ⎤
⎣ f ( x )− g ( x ) ⎤
⎦ dx .
Then,
b
ρ ∫
x ⎡ ⎣ f ( x )− g ( x ) ⎤
⎦ dx
a
d =2 π
a b
ρ ∫
⎡ ⎣ f ( x )− g ( x ) ⎤
⎦ dx
and thus
d · A =2 π ∫ a b
x ⎡ ⎣ f ( x )− g ( x ) ⎤
⎦ dx .
However, using the method of cylindrical shells, we have V =2 π ∫ a b x ⎡
⎣ f ( x )− g ( x ) ⎤
⎦ dx .
So,
V = d · A
and the proof is complete. □ Example 6.34 Using the Theorem of Pappus for Volume
Let R be a circle of radius 2 centered at (4, 0). Use the theorem of Pappus for volume to find the volume of the torus generated by revolving R around the y -axis.
Made with FlippingBook - professional solution for displaying marketing and sales documents online