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Mathematics
Problem
Which integral represents the exact length of a smooth curve given by
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
on the interval
[
a
,
b
]
[a,b]
[
a
,
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
+
(
d
x
d
y
)
2
d
y
\displaystyle\int_{a}^{b}\sqrt{1+\bigl(\tfrac{dx}{dy}\bigr)^{2}}\,dy
∫
a
b
1
+
(
d
y
d
x
)
2
d
y
c)
∫
a
b
1
+
(
f
′
(
x
)
)
2
d
x
\displaystyle\int_{a}^{b}\sqrt{1+\bigl(f'(x)\bigr)^{2}}\,dx
∫
a
b
1
+
(
f
′
(
x
)
)
2
d
x
d)
∫
a
b
(
f
′
(
x
)
)
2
d
x
\displaystyle\int_{a}^{b}\sqrt{\bigl(f'(x)\bigr)^{2}}\,dx
∫
a
b
(
f
′
(
x
)
)
2
d
x
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