Part II: The Answers
386
Answers
601–700
635.
4 2
3 1
2
−
+
Begin by finding the velocity function by evaluating the antiderivative of the acceleration function, a
 
(t) = sin t + cos t:
v t
t
t dt
t
t C
( )
(sin cos )
cos sin
=
+
= −
+
+
∫
Next, use the initial condition, v π
4
0
( ) = , to solve for the arbitrary constant of
integration:
C
v
C
C
C
π
π
π
π
π
4
4
4
0
4
4
0
2
2
2
2
0
( ) = − + +
= −
+
+
= −
+
+
=
cos
sin
cos
sin
Therefore, the velocity function is
v t
t
t
( )
cos sin
= −
+
To find the distance traveled, integrate the absolute value of the velocity function over
the given interval, π
π
6
≤ ≤
t
:
−
+
∫ cos sin
t
t dt
π
π
6
Find any zeros of the velocity function on the given interval so you can determine
where the velocity function is positive or negative:
−
+
=
=
=
cos sin
sin
cos
t
t
t
t
t
0
4
π
The velocity function is negative on the interval π π
6 4
,
( ) and positive on the interval
π π
4
,
( ) , so the total distance traveled is
− −
+
(
) +
−
+
(
)
=
+
(
) + −
∫
∫
cos sin
cos sin
sin cost
s
t
t dt
t
t dt
t
π
π
π
π
π
π
4
6
4
6
4
i in cos
sin
cos
sin
cos
sin
cos
s
t
t
−
(
)
=
+
(
) − +
(
) + − −
(
) − −
π
π
π
π
π
π
π
π
4
4
4
6
6
i in
cos
π
π
4
4
4 2
3 1
2
−
(
)

 

 
=
−
+