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Mathematics
Mathematics
Problem
Which of the following definite integrals gives the exact arc length of the graph of a differentiable function
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+\bigl(f'(x)\bigr)^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+\bigl(f(x)\bigr)^2}\,dx
∫
a
b
1
+
(
f
(
x
)
)
2
d
x
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
)
∣
d
x
\displaystyle \int_a^b \lvert f'(x)\rvert\,dx
∫
a
b
∣
f
′
(
x
)∣
d
x
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