Suppose that we are given a function
f
{\displaystyle f}
that is continuous on an interval
[
a
,
b
]
{\displaystyle [a,b]}
and we want to calculate the length of the curve drawn out by the graph of
f
(
x
)
{\displaystyle f(x)}
from
x
=
a
{\displaystyle x=a}
to
x
=
b
{\displaystyle x=b}
. If the graph were a straight line this would be easy — the formula for the length of the line is given by Pythagoras' theorem. And if the graph were a piecewise linear function we can calculate the length by adding up the length of each piece.
The problem is that most graphs are not linear. Nevertheless we can estimate the length of the curve by approximating it with straight lines. Suppose the curve
C
{\displaystyle C}
is given by the formula
y
=
f
(
x
)
{\displaystyle y=f(x)}
for
a
≤
x
≤
b
{\displaystyle a\leq x\leq b}
. We divide the interval
[
a
,
b
]
{\displaystyle [a,b]}
into
n
{\displaystyle n}
subintervals with equal width
Δ
x
{\displaystyle \Delta x}
and endpoints
x
0
,
x
1
,
…
,
x
n
{\displaystyle x_{0},x_{1},\ldots ,x_{n}}
. Now let
y
i
=
f
(
x
i
)
{\displaystyle y_{i}=f(x_{i})}
so
P
i
=
(
x
i
,
y
i
)
{\displaystyle P_{i}=(x_{i},y_{i})}
is the point on the curve above
x
i
{\displaystyle x_{i}}
. The length of the straight line between
P
i
{\displaystyle P_{i}}
and
P
i
+
1
{\displaystyle P_{i+1}}
is
|
P
i
P
i
+
1
|
=
(
y
i
+
1
−
y
i
)
2
+
(
x
i
+
1
−
x
i
)
2
{\displaystyle {\bigl |}P_{i}P_{i+1}{\bigr |}={\sqrt {(y_{i+1}-y_{i})^{2}+(x_{i+1}-x_{i})^{2}}}}
So an estimate of the length of the curve
C
{\displaystyle C}
is the sum
∑
i
=
0
n
−
1
|
P
i
P
i
+
1
|
{\displaystyle \sum _{i=0}^{n-1}{\bigl |}P_{i}P_{i+1}{\bigr |}}
As we divide the interval
[
a
,
b
]
{\displaystyle [a,b]}
into more pieces this gives a better estimate for the length of
C
{\displaystyle C}
. In fact we make that a definition.
Arclength of a parametric curve
edit
For a parametric curve, that is, a curve defined by
x
=
f
(
t
)
{\displaystyle x=f(t)}
and
y
=
g
(
t
)
{\displaystyle y=g(t)}
, the formula is slightly different:
L
=
∫
a
b
f
′
(
t
)
2
+
g
′
(
t
)
2
d
t
{\displaystyle L=\int \limits _{a}^{b}{\sqrt {f'(t)^{2}+g'(t)^{2}}}\,dt}
Proof: The proof is analogous to the previous one:
Consider
y
i
+
1
−
y
i
=
g
(
t
i
+
1
)
−
g
(
t
i
)
{\displaystyle y_{i+1}-y_{i}=g(t_{i+1})-g(t_{i})}
and
x
i
+
1
−
x
i
=
f
(
t
i
+
1
)
−
f
(
t
i
)
{\displaystyle x_{i+1}-x_{i}=f(t_{i+1})-f(t_{i})}
.
By the Mean Value Theorem there are points
c
i
{\displaystyle c_{i}}
and
d
i
{\displaystyle d_{i}}
in
(
t
i
+
1
,
t
i
)
{\displaystyle (t_{i+1},t_{i})}
such that
y
i
+
1
−
y
i
=
g
(
t
i
+
1
)
−
g
(
t
i
)
=
g
′
(
c
i
)
(
t
i
+
1
−
t
i
)
{\displaystyle y_{i+1}-y_{i}=g(t_{i+1})-g(t_{i})=g'(c_{i})(t_{i+1}-t_{i})}
and
x
i
+
1
−
x
i
=
f
(
t
i
+
1
)
−
f
(
t
i
)
=
f
′
(
d
i
)
(
t
i
+
1
−
t
i
)
{\displaystyle x_{i+1}-x_{i}=f(t_{i+1})-f(t_{i})=f'(d_{i})(t_{i+1}-t_{i})}
So
|
P
i
P
i
+
1
|
{\displaystyle {\bigl |}P_{i}P_{i+1}{\bigr |}}
=
(
x
i
+
1
−
x
i
)
2
+
(
y
i
+
1
−
y
i
)
2
{\displaystyle ={\sqrt {(x_{i+1}-x_{i})^{2}+(y_{i+1}-y_{i})^{2}}}}
=
f
′
(
d
i
)
2
(
t
i
+
1
−
t
i
)
2
+
g
′
(
c
i
)
2
(
t
i
+
1
−
t
i
)
2
{\displaystyle ={\sqrt {f'(d_{i})^{2}(t_{i+1}-t_{i})^{2}+g'(c_{i})^{2}(t_{i+1}-t_{i})^{2}}}}
=
(
f
′
(
d
i
)
2
+
g
′
(
c
i
)
2
)
(
t
i
+
1
−
t
i
)
2
{\displaystyle ={\sqrt {{\bigl (}f'(d_{i})^{2}+g'(c_{i})^{2}{\bigr )}(t_{i+1}-t_{i})^{2}}}}
=
f
′
(
d
i
)
2
+
g
′
(
c
i
)
2
Δ
t
{\displaystyle ={\sqrt {f'(d_{i})^{2}+g'(c_{i})^{2}}}\Delta t}
Putting this into the definition of the length of the curve gives
L
=
lim
n
→
∞
∑
i
=
0
n
−
1
f
′
(
d
i
)
2
+
g
′
(
c
i
)
2
Δ
t
{\displaystyle L=\lim _{n\to \infty }\sum _{i=0}^{n-1}{\sqrt {f'(d_{i})^{2}+g'(c_{i})^{2}}}\Delta t}
This is equivalent to:
L
=
∫
a
b
f
′
(
t
)
2
+
g
′
(
t
)
2
d
t
{\displaystyle L=\int \limits _{a}^{b}{\sqrt {f'(t)^{2}+g'(t)^{2}}}\,dt}
3. Find the circumference of the circle given by the parametric equations
x
(
t
)
=
R
cos
(
t
)
{\displaystyle x(t)=R\cos(t)}
,
y
(
t
)
=
R
sin
(
t
)
{\displaystyle y(t)=R\sin(t)}
, with
t
{\displaystyle t}
running from
0
{\displaystyle 0}
to
2
π
{\displaystyle 2\pi }
.
:
2
π
R
{\displaystyle 2\pi R}
:
2
π
R
{\displaystyle 2\pi R}
4. Find the length of one arch of the
cycloid given by the parametric equations
x
(
t
)
=
R
(
t
−
sin
(
t
)
)
{\displaystyle x(t)=R{\bigl (}t-\sin(t){\bigr )}}
,
y
(
t
)
=
R
(
1
−
cos
(
t
)
)
{\displaystyle y(t)=R{\bigl (}1-\cos(t){\bigr )}}
, with
t
{\displaystyle t}
running from
0
{\displaystyle 0}
to
2
π
{\displaystyle 2\pi }
.
:
8
R
{\displaystyle 8R}
:
8
R
{\displaystyle 8R}
Solutions