Question

Fill in the blanks to complete each of the following theorem:

Let C be a curve in the plane with parametrization,$y=g\left(t\right)$ for $t\in \left[a,b\right]$such that the parametrization is a................... function from the interval $\left[a,b\right]$to the curve C. If $x=f\left(t\right)$and $y=g\left(t\right)$are differentiable functions of t such that $f\text{'}\left(t\right)$and $g\text{'}\left(t\right)$are .............on $\left[a,b\right]$, then the length of the curve C is given by...................

Short answer

The blanks can be filled in order as:

1. One- to-one

2. Continuous

3.${\int }_a^b\sqrt{{\left(f\text{'}\left(t\right)\right)}^2+{\left(g\text{'}\left(t\right)\right)}^2}dt$

Step-by-step solution

Step 1. Given information

C is the curve

$$\begin{gathered} x=f\left(t\right) \\ y=g\left(t\right) \end{gathered}$$

for $t\in [a,b]$

$f\text{'}\left(t\right),g\text{'}\left(t\right)$are continuous over the interval.

Step 2. Write formula for length of the curve.

The Length of the curve in the given interval is:

$l={\int }_a^b\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}$

Since, $x=f\left(t\right)$, hence $\frac{dx}{dt}=f\text{'}\left(t\right)$

$$\begin{gathered} y=g\left(t\right) \\ \frac{dy}{dt}=g\text{'}\left(t\right) \end{gathered}$$

So the length of the curve is:

$l={\int }_a^b\sqrt{{\left(f\text{'}\left(t\right)\right)}^2+{\left(g\text{'}\left(t\right)\right)}^2}dt$