Find the exact length of the curve.
x = e^t + e^−t, y = 5 − 2t, 0 ≤ t ≤ 4
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Answer:
L
=
∫
3
0
√
(
e
t
−
e
−
t
)
2
+
(
−
2
)
2
d
t
=
∫
3
0
√
e
2
t
−
2
+
e
−
2
t
+
4
d
t
expand
=
∫
3
0
√
e
2
t
+
2
+
e
−
2
t
d
t
simplify
=
∫
3
0
√
(
e
t
+
e
−
t
)
2
d
t
factor
=
∫
3
0
(
e
t
+
e
−
t
)
d
t
simplify
=
e
t
−
e
−
t
∣
∣
3
0
integrate
=
e
3
−
e
−
3
−
(
e
0
−
e
0
)
evaluate
=
e
3
−
e
−
3
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