Length Of Any Curve
Let’s suppose is the equation
of any curve. Then the length of the curve between
and
is
According to Stroud and Booth (2013)*, “Find
the length of the arc of the curve , between
and
.”
Now here the equation of the curve is
(1)
First of all, I’ll get the value of .
So, from equation (1), I can say that
the value of is
Next, I’ll differentiate with respect
to
to
get
Then I’ll simplify it to get
So this means
Therefore the value of is
If I simplify it, I’ll get
Next, I’ll get the value of .
At first, I’ll get the value of .
Thus it will be
Now I’ll simplify it to get
So this gives
Therefore the value of is
And that means
Now I’ll get the length of the arc of
the curve
, between
and
.
So it will be
And that means
Next, I’ll integrate it to get
Then I’ll substitute the limits to get
Now I’ll simplify it. And that gives
Thus the length of the arc is . Hence I can conclude that this is the answer to the
given example.
Now I’ll give another example.
According to Stroud and Booth (2013)*, “Find
the length of the curve between
and
.”
Now here the equation of the curve is
(2)
First of all, I’ll get the value of .
So I’ll differentiate equation (2)
throughout with respect to . And that gives
Next, I’ll simplify it to get
So this means
And that gives the value of as
Now I’ll get the value of . So that will be
Next, I’ll get the value of .
At first, I’ll get the value of .
Thus it will be
Now I’ll simplify it to get
So this gives
Therefore the value of is
And that means
Now I’ll get the the length of the curve between
and
.
So it will be
And that means
Next, I’ll integrate it to get
Then I’ll substitute the limits to get
Now I’ll simplify it and that gives
Thus the length of the curve is .
Hence I can conclude that this is the answer to the given example.