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Mathematics
Problem
For a smooth curve defined by
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
on
[
a
,
b
]
[a,b]
[
a
,
b
]
, which of the following definite integrals gives the length of the curve between
x
=
a
x=a
x
=
a
and
x
=
b
x=b
x
=
b
? a)
∫
a
b
(
1
+
[
f
′
(
x
)
]
2
)
d
x
\displaystyle \int_{a}^{b} \bigl(1+[f'(x)]^{2}\bigr)\,dx
∫
a
b
(
1
+
[
f
′
(
x
)
]
2
)
d
x
b)
∫
a
b
1
+
[
f
′
(
x
)
]
2
d
x
\displaystyle \int_{a}^{b} \sqrt{1+[f'(x)]^{2}}\,dx
∫
a
b
1
+
[
f
′
(
x
)
]
2
d
x
c)
∫
a
b
∣
f
′
(
x
)
∣
d
x
\displaystyle \int_{a}^{b} |f'(x)|\,dx
∫
a
b
∣
f
′
(
x
)
∣
d
x
d)
∫
a
b
1
+
[
f
(
x
)
]
2
d
x
\displaystyle \int_{a}^{b} \sqrt{1+[f(x)]^{2}}\,dx
∫
a
b
1
+
[
f
(
x
)
]
2
d
x
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