374
Chapter Seven INTEGRATION
Substitution
Getting an integrand into the right form to use a table of integrals involves substitution and a variety
of algebraic techniques.
et sin(5t + 7) dt.
Example 9
Find
Solution
This looks similar to II-8. To make the
8.2 APPLICATIONS TO GEOMETRY
427
Arc Length of a Parametric Curve
A particle moving along a curve in the plane given by the parametric equations x = f (t), y = g(t),
where t is time, has speed given by:
!2
!2
dx
dy
v(t) =
+
.
dt
dt
We can find the distanc
4.7 LHOPITALS RULE, GROWTH, AND DOMINANCE
247
(d) We have
1
1
1 1
= = .
x sin x
0 0
Limits like this can often be calculated by adding the fractions:
sin x x
1
1
= lim
,
lim
x0
x0 x
sin x
x sin x
giving a 0/0 form that can be evaluated using lHopitals ru
360
Chapter Seven INTEGRATION
Exercises and Problems for Section 7.1
Exercises
1. Use substitution to express each of the following inteb
grals as a multiple of a (1/w) dw for some a and b.
Then evaluate the integrals.
1
x
dx
1 + x2
(a)
0
/4
(b)
0
sin x
d
368
Chapter Seven INTEGRATION
On the right side we have an integral similar to the original one, with the sine replaced by a cosine.
Using integration by parts on that integral in the same way gives
1
2
e2x sin(3x) dx.
e2x cos(3x) dx = e2x sin(3x)
3
3
Su
7.6 IMPROPER INTEGRALS
6
0
401
1
dx = 3(4)1/3 + 3(2)1/3 = 8.542.
(x 4)2/3
Finally, there is a question of what to do when an integral is improper at both endpoints. In this
case, we just break the integral at any interior point of the interval. The origin
8.1 AREAS AND VOLUMES
419
Exercises and Problems for Section 8.1
Exercises
y
1. (a) Write a Riemann sum approximating the area of the
region in Figure 8.13, using vertical strips as shown.
(b) Evaluate the corresponding integral.
y = x2 + 6x
9
y
y
y = 2x
384
Chapter Seven INTEGRATION
Since x + 1/2 = ( 3/2) tan , we have = arctan(2/ 3)x + 1/ 3), so
!
1
2
1
2
+ C.
dx = arctan x +
x2 + x + 1
3
3
3
Alternatively, using a computer algebra system gives7
2 tan
1
!
2x + 1
3
,
3
essentially the same as we obtaine