9 FURTHER APPLICATIONS OF INTEGRATION

SECTION 9.1
ARC LENGTH
SPECIAL PROJECT: How to Define π
This project allows the students to deduce a historic geometric result using techniques from this section.
They start by defining π as the circumference of a semi-circle of radius 1, and then show that the integral
representing this quantity is the same as the integral representing the area of a circle of radius 1.
Note that the simplification in Problem 3 requires an algebraic trick. The students may need the hint that
∫
x 2 ∫ x 2
√ dx = − 1 + 1
√ dx. They will also need to be able to integrate by parts.
1 − x 2 1 − x 2
Answers:
√
∫ 1
1. 1 +
−1
[ d
(√
1 − x 2) ] 2 ∫ 1
dx =
dx
−1
dx
√
1 − x 2
2. The term on the left is the length of a semicircle of radius 1, and the term on the right is an expression of
the area of a circle of radius 1.
3. I = ∫ √ 1 − x 2 dx.Letu = √ 1 − x 2 , dv = dx ⇒ du =−√ xdx , v = x. Then
1 − x 2
√ ∫
I = x 1 − x 2 +
x 2 ∫
dx
√
√1 = x − x 2 +
1 − x 2
dx
√
1 − x 2 − ∫ √
1 − x 2 dx
4. Using the answer to Problem 3, 2 ∫ 1
√
−1 1 − x 2 dx = x √ ∣ ∫
1 − x 2 ∣∣ 1 1
+
−1
−1
dx
√
1 − x 2
and the result follows.
HOMEWORK PROBLEMS
Core Exercises: 3, 8, 13, 17, 19, 22, 26, 31, 34, 39
Sample Assignment: 2, 3, 6, 8, 10, 13, 17, 19, 22, 25, 26, 27, 31, 34, 37, 39, 40
Exercise D A N G
2 ×
3 ×
6 ×
8 ×
10 ×
13 ×
17 ×
19 ×
22 × ×
25 ×
26 ×
27 × ×
31 × ×
34 × ×
37 × ×
39 × ×
40 × ×
487