The Arc Length of a Parabola - Drexel University
Integral Calculus Grinshpan
The Arc Length of a Parabola Let us calculate the length of the parabolic arc y = x2, 0 x a.
According to the arc length formula,
a
a
L(a) =
1 + y (x)2 dx =
1 + (2x)2 dx.
0
0
1 Replacing 2x by x, we may write L(a) =
2a 1 + x2 dx.
20
Thus the task is to find the antiderivative of 1 + x2.
This is often done by setting x = sinh t or x = tan t.
We will obtain the answer by manipulating square roots.
Two observations are needed:
x x2 + 1
=
x2
+1+
x2
x2 + 1
= x2 + 1 + x2 + 1 -
1
x2 + 1
= 2 x2 + 1 -
1
.
x2 + 1
ln x + 1 + x2
1 + x/ 1 + x2
=
x + 1 + x2
1
=
.
1 + x2
Hence 2 1 + x2 = x x2 + 1 + ln x + 1 + x2 .
Hence
1
+
x2
dx
=
1
x 1+
x2
+
1
ln
x + 1 + x2
+ C.
2
2
1
1
It follows that L(a) = a 1 + 4a2 + ln 2a + 1 + 4a2 .
2
4
1 1
In particular, L(1) = 5 + ln(2 + 5) 1.48.
2
4
................
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