The length of the curve given by the parametric equations x = 2sin(3t) and y = 2cos(3t) is 12 units.
Explanation: To find the length of the curve, we can use the arc length formula for parametric curves:
�
=
∫
�
�
(
�
�
�
�
)
2
+
(
�
�
�
�
)
2
�
�
s=∫
a
b
(
dt
dx
)
2
+(
dt
dy
)
2
dt
where
�
s is the length of the curve and
�
t ranges from
�
a to
�
b.
In this case,
�
=
2
sin
(
3
�
)
x=2sin(3t) and
�
=
2
cos
(
3
�
)
y=2cos(3t), so we need to find
�
�
�
�
dt
dx
and
�
�
�
�
dt
dy
:
�
�
�
�
=
2
⋅
3
cos
(
3
�
)
=
6
cos
(
3
�
)
dt
dx
=2⋅3cos(3t)=6cos(3t)
�
�
�
�
=
−
2
⋅
3
sin
(
3
�
)
=
−
6
sin
(
3
�
)
dt
dy
=−2⋅3sin(3t)=−6sin(3t)
Now, we can plug these values back into the arc length formula and integrate:
�
=
∫
�
�
(
6
cos
(
3
�
)
)
2
+
(
−
6
sin
(
3
�
)
)
2
�
�
s=∫
a
b
(6cos(3t))
2
+(−6sin(3t))
2
dt
�
=
∫
�
�
36
cos
2
(
3
�
)
+
36
sin
2
(
3
�
)
�
�
s=∫
a
b
36cos
2
(3t)+36sin
2
(3t)
dt
�
=
∫
�
�
36
(
cos
2
(
3
�
)
+
sin
2
(
3
�
)
)
�
�
s=∫
a
b
36(cos
2
(3t)+sin
2
(3t))
dt
�
=
∫
�
�
36
�
�
s=∫
a
b
36
dt
�
=
6
�
∣
�
�
s=6t
∣
∣
a
b
Conclusion: The length of the curve is
6
�
−
6
�
6b−6a. As we don't have the range of
�
t specified (from
�
a to
�
b), we cannot determine the exact length. However, if the range is from
�
=
0
t=0 to
�
=
2
�
t=2π (one full revolution), then
�
=
2
�
b=2π and
�
=
0
a=0, and the length would be
6
(
2
�
)
−
6
(
0
)
=
12
6(2π)−6(0)=12.