Page 243 - Calculus Workbook For Dummies

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Chapter 12: Who Needs Freud? Using the Integral to Solve Your Problems

Arc Length and Surfaces of Revolution

Integration determines the length of a curve by cutting it up into an infinite number of
infinitesimal segments, each of which is sort of the hypotenuse of a tiny right triangle.
Then your pedestrian Pythagorean Theorem does the rest. The same basic idea applies
to surfaces of revolution. Here are two handy formulas for solving these problems:
Arc length: The length along a function, f xh, from a to b is given by
^
b
Arc Length= # 1 + ` f l ^ hj 2 dx
x
a
Surface of revolution: The surface area generated by revolving the portion of a
function, f xh, between x = a and x = b about the x-axis is given by
^
b 2
π
=
x
f x
Surface Area 2 # ^ h 1 + ` f l ^ hj dx
a
Q. What’s the arc length along f x = x / 2 3 from Q. Find the surface area generated by revolving
^ h
x = 8 to x = 27? 1 3
f x = x ^ 0 # x # 2h about the x-axis.
^ h
3
A. The arc length is about 19.65. A. π
The area is `
9 17 17 - 1j.
1. Find f l ^ xh.
1. Find the function’s derivative.
2 - / 1 3
/ 2 3
x =
f x = x f l ^ h x 1 3 2
^ h
x =
^ h
3 f x = x f l ^ h x
3
2. Plug into the arc length formula.
2. Plug into the surface area formula.
27
27 = # 4 - / 2 3 2 1 2
π
2
=
Arc Length to8 1 + 9 x dx Surface Area 2 # x 3 1 + _ x i dx
8 3
0
3. Integrate. π 2
4
= 2 x # 3 1 + x dx
These arc length problems tend to pro- 3
0
duce tricky integrals; I’m not going to
show all the work here. You can do this integral with
u-substitution.
27
1
= # 9 + 4 x - / 2 3 dx
3 u 1 + x 4 when x = 0 , u 1
=
=
8
3
=
=
27 du 4 x dx when x = 2 , u 17
1
/ 2 3
= x # - / 1 3 9 x + 4 dx
3 2
8 2 π 1
4
= $ 4 # x 3 1 + x dx
You finish this with a u-substitution, 3 4 0
/ 2 3
where u 9= x + 4. 17
π
= # / 1 2
85 u du
1
= # 1 u du 6 1
/ 1 2
3 6 17
40 π 2 / 3 2
85 = ; u E
1 2 / 3 2 6 3
= ; u E 1
18 3 π
40
= ` 17 17 - 1j
85 85 - 80 10 9
=
27
. 19 .65
An eminently sensible answer, because
from x = 8 to x = 27, x / 2 3 is very similar to
a straight line of length 27 – 8, which
equals 19.