375
Answers
601–700
Answers and Explanations
2
2
1
2
2
2
2
2
2
cot
csc
csc
c sc
csc
cot
x
x dx
x
xdx
x
dx
x
−
(
)
=
−
(
) −
(
)
=
−
(
)
= −
∫
∫
∫
− −
+
2x C
610.
sin x + C
Begin by factoring cos x from the numerator and use a trigonometric identity to
simplify:
cos
cos tan
sec
cos
tan
sec
cos sec
se
x
x
x
x
dx
x
x
x
dx
x
x
+
=
+
(
)
=
(
)
∫
∫
2
2
2
2
2
1
c c
cos
sin
2 x
dx
xdx
x C
∫
∫
=
=
+
611.
the net change in the baby’s weight in pounds during the first 2 weeks of life
The net change theorem states that the integral of a rate of change is the net change:
′
=
−
∫ F x dx F b F a
a
b
( )
( )
( )
Because w'(t) is the rate of the baby’s growth in pounds per week, w(t) represents the
child’s weight in pounds at age t. From the net change theorem, you have
′
=
−
∫ w t dt w
w
( )
( )
( )
0
2
2
0
Therefore, the integral represents the increase in the child’s weight (in pounds) from
birth to the start of the second week.
612.
the total amount of oil, in gallons, that leaks from the tanker for the first 180 minutes after
the leak begins.
Because r(t) is the rate at which oil leaks from a tanker in gallons per minute, you can
write r(t) = –V'(t), where V(t) is the volume of the oil at time t. Notice that you need the
minus sign because the volume is decreasing, so V'(t) is negative but r(t) is positive. By
the net change theorem, you have
r t dt
V t dt
V
V
V
V
( )
( )
(
)
( )
( )
(
)
0
180
0
180
180
0
0
180
∫
∫
= −
′
= −
−
[
]
=
−
This is the number of gallons of oil that leaks from the tank in the first 180 minutes
after the leak begins.