# What is the arclength of #f(t) = (sqrt(t^2-t^3),t^3-t^2)# on #t in [-1,1]#?

##### 2 Answers

#### Explanation:

We have

then

by the chain rule

then

by the power rule

so we have to solve

we get by a numerical method

#### Explanation:

It is easy to see that the curve traced out in this case is part of a parabola

Thus, the infinitesimal arc-length between two neighboring points on this curve is given simply by

The only real problem in calculating the total arc length is that as

graph{sqrt(x^2-x^3) [-1.1, 1.1, -0.5, 1.5]}

As can be seen clearly, the value of

- changes monotonously from
#sqrt 2# to 0 as#t# goes from -1 to 0. - After this, it increases from 0 to some
#0 < x _0 < 1# as#t# increases from 0 to some#0< t_0< 1# - and then returns from
#x_0# back to 0 as#t# goes from#t_0# to 1.

It is easy to see that

So the parabola

- from
#x=sqrt 2# to#x=0#

arc length#L_1=| int_sqrt2^0 sqrt{1+4x^2}dx|# - from
#x=0# to#x=x_0=sqrt{4/27}#

arc length#L_2=| int_0^{sqrt{4/27}} sqrt{1+4x^2}dx|# - from
#x=x_0=sqrt{4/27}# back to#x=0#

arc length#L_3=L_2#

Since

we have

Thus the total arc length is