Q. 52 - Calculus

Question

In Exercises 49-53 sketch the parametric curve and find its length.

$x=e^t\cos t,y=e^t\sin t,t\in [0,1]$

Step-by-Step Solution

Answer

The graphical representation of the points $(1,0)(2.71,0.01)$ is as follows,

Therefore, the length of the curve is equals to $\sqrt{2}[e-1] or 2.43$ .

Step 1: Given information

The parametric curve $x=e^t\cos t,y=e^t\sin t,t\in [0,1]$

Step 2: Calculation

Consider the parametric equations $x=e^{\text{'}}\cos t,y=e^{\text{'}}\sin t,t\in [0,1]$.

The objective is to draw the parametric curve and find the arc length of the curve.

The formula to find the arc length of the curve is ,

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

First find the derivative of the parametric equations $x=e^{\text{'}}\cos t,y=e^{\text{'}}\sin t$.

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That is $f\left(t\right)=e^t\cos t$

The derivative with respect to t is written as follows,

$f^{\text{'}}\left(t\right)=\frac{d}{dt}\left(e^{\text{'}}\cos t\right)$

On simplifying.

$$\begin{gathered} f^{\text{'}}\left(t\right)=e^{\text{'}}\frac{d}{dt}\left(\cos t\right)+\cos t\frac{d}{dt}\left(e^{\text{'}}\right)[sinced(uv)=udv+vdu] \\ {f}^{\text{'}}\left(t\right)=e^{\text{'}}(-\sin t)+\cos t\left(e^{\text{'}}\right) \\ {f}^{\text{'}}\left(t\right)=-e^{\text{'}}\sin t+e^{\text{'}}\cos t \end{gathered}$$

Step 3: Simplifying

Take $y=e^{\text{'}}\sin t$

That is $g\left(t\right)=e^{\text{'}}\sin t$

The derivative with respect to t is written as follows,

$$\begin{gathered} \frac{d}{dt}g\left(t\right)=\frac{d}{dt}\left(e^{\text{'}}\sin t\right) \\ \frac{d}{dt}{g}^{\text{'}}\left(t\right)=e^{\text{'}}\frac{d}{dt}\left(\sin t\right)+\sin t\frac{d}{dt}\left(e^{\text{'}}\right)[sinced(uv)=udv+vdu] \end{gathered}$$

Thus,

$g^{\text{'}}\left(t\right)=e^{\text{'}}\cos t+\sin t{e}^{\text{'}}$

Now by using the, Length of the curve $={\int }_a^b\sqrt{{\left(f^{\text{'}}\left(t\right)\right)}^2+{\left(g^{\text{'}}\left(t\right)\right)}^2}dt$

Length of the curve $={\int }_0^1\sqrt{{\left(-e^{\text{'}}\sin t+e^{\text{'}}\cos t\right)}^2+{\left(e^{\text{'}}\cos t+\sin t{e}^{\text{'}}\right)}^2}dt$

$={\int }_0^1\sqrt{e^{2t}(-\sin t+\cos t{)}^2+e^{2t}(\cos t+\sin t{)}^2}dt$

Step 4: Further simplification

On further simplification,

$={\int }_0^1{e}^{\text{'}}\sqrt{\left({\sin }^{2}t+{\cos }^{2}t-2\cos t\sin t\right)+\left({\cos }^{2}t+{\sin }^{2}t+2\cos t\sin t\right)}dt$ $={\int }_0^1{e}^{\text{'}}\sqrt{(1-2\cos t\sin t)+(1+2\cos t\sin t)}dt$ 
$={\int }_0^1{e}^{\text{'}}\sqrt{1-2\cos t\sin t+1+2\cos t\sin t}dt$

Length of the curve$={\int }_0^1{e}^{\text{'}}\sqrt{2}dt$

Thus,

Length of the curve $={\sqrt{2}}^1{\int }^{\text{'}}dt$

$=\sqrt{2}{\left[e^{\text{'}}\right]}_0^1$

By substituting the limits,

Length of the curve $=\sqrt{2}\left[e^3-e^0\right]$

Length of the curve $=\sqrt{2}[e-1]=2.43$

To draw the curve for the parametric equations, first find the points when $t\in [0,1]$.

Given that $t\in [0,1]$

When $t=0$.

$$\begin{gathered} (x,y)=\left(e^{\text{'}}\cos t,e^{\text{'}}\sin t\right) \\ (x,y)=\left(e^0\cos 0,e^0\sin 0\right) \\ (x,y)=(1,0) \end{gathered}$$

When t=1.

$(x,y)=\left(e^{\text{'}}\cos t,e^{\text{'}}\sin t\right)$

Then,

$(x, y)=(2.71,0.01)$

Step 5: Plot the graph

The graphical representation of the points $(1,0)(2.71,0.01)$ is as follows,

Therefore, the length of the curve is equals to $\sqrt{2}[e-1] or 2.43.$