Calculus- Early Transcendentals, 2021a

330 Applications of Integration
Unfortunately, integrals of this form are typically difficult or impossible to compute exactly, because
usually none of our methods for finding antiderivatives will work. In practice this means that the integral
will usually have to be approximated.
Example 8.22: Circumference of a Circle
√
Let f (x)= r 2 − x 2 , the upper half circle of radius r. The length of this curve is half the circumference,
namely πr. Compute this with the arc length formula.
√
Solution. The derivative f ′ is −x/ r 2 − x 2 so the integral is
√
√
∫ r
1 + x2 ∫ r
−r r 2 − x 2 dx = r 2 ∫ √ r 1
−r r 2 − x 2 dx = r −r r 2 − x 2 dx.
Using a trigonometric substitution, we find the antiderivative,
√
namely arcsin(x/r). Notice that the integral
is improper at both endpoints, as the function 1/(r 2 − x 2 ) is undefined when x = ±r. So we need to
compute
∫ √ 0
∫ √
1
D 1
lim
D→−r + D r 2 dx+ lim
− x2 D→r − 0 r 2 − x 2 dx.
This is not difficult, and has value π, so the original integral, with the extra r in front, has value πr as
expected.
♣
Exercises for 8.7
Exercise 8.7.1 Find the arc length of f (x)=x 3/2 on [0,2].
Exercise 8.7.2 Find the arc length of f (x)=x 2 /8 − lnxon[1,2].
Exercise 8.7.3 Find the arc length of f (x)=(1/3)(x 2 + 2) 3/2 on the interval [0,a].
Exercise 8.7.4 Find the arc length of f (x)=ln(sinx) on the interval [π/4,π/3].
Exercise 8.7.5 Let a > 0. Show that the length of y = coshxon[0,a] is equal to
Exercise 8.7.6 Find the arc length of f (x)=coshxon[0,ln2].
∫ a
0
coshxdx.
Exercise 8.7.7 Set up the integral to find the arc length of sinx on the interval [0,π]; do not evaluate the
integral. If you have access to appropriate software, approximate the value of the integral.
Exercise 8.7.8 Set up the integral to find the arc length of y = xe −x on the interval [2,3]; do not evaluate
the integral. If you have access to appropriate software, approximate the value of the integral.
Exercise 8.7.9 Find the arc length of y = e x on the interval [0,1]. (This can be done exactly; it is a bit
tricky and a bit long.)
Find the arc length of the function on the given interval.