Answer:
L=32.32
Step-by-step explanation:
In order to solve this problem, we must first set the integral we need to solve. For this matter, we will need to make use of the Length of parametric functions formula, which looks like this:
[tex]L=\int\limits^a_b {\sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}}} \, dt[/tex]
So first, we need to find what the derivative of each parameter is:
we get: [tex]x=t-e^{t}[/tex]
and its derivative is:
[tex]\frac{dx}{dt}=1-e^{t}[/tex]
we also have that:
[tex]y=t+e^{t}[/tex]
and its derivative is:
[tex]\frac{dy}{dt}=1+e^{t}[/tex]
So we can use these derivatives into our length formula, so we get:
[tex]L=\int\limits^{-3}_{3} {\sqrt{(1-e^{t})^{2}+(1+e^{t})^{2}}} \, dt[/tex]
this integral we got here can be simplified by expanding each parenthesis, for which we get:
[tex]L=\int\limits^{-3}_{3} {\sqrt{1-2e^{t}+e^{2t}+1+2e^{t}+e^{2t}}} \, dt[/tex]
We can now combine like terms so we get:
[tex]L=\int\limits^{-3}_{3} {\sqrt{2+2e^{2t}}} \, dt[/tex]
So we can now use Simpson's rule to approximate this integral.
Simpson's rule general formula looks like this:
[tex]\int\limits^{b}_{a}{f(x)}\, dx=\frac{\Delta x}{3}[f(x_{0})+4f(x_{1})+2f(x_{2})+...+2f(x_{n-2})+4f(x_{n-1})+f(x_{n})][/tex]
The idea of this rule is to keep the fist and last result of the functions the same, multiply the even terms by 2 and the odd terms by 4.
So we start by finding [tex]\Delta x[/tex]
[tex]\Delta x=\frac{b-a}{n}[/tex]
in this case a=-3, b=3 and n=6 so we get:
[tex]\Delta x=\frac{3-(-3)}{6}=1[/tex]
So with this we can build a table we can use to find the values used in the Simpson's rule. See uploaded table.
Each of the values were found by substituting each of the x-values into the functions we are integrating. Take for example x=-3:
[tex]f(-3)=\sqrt{2+2e^{2(-3)}}=1.41596522[/tex]
so now we can take each of those values and plug them into the simpson's rule so we get:
[tex]L=\int\limits^{-3}_{3} {\sqrt{2+2e^{2t}}} \, dt=\frac{1}{3}[1.416+4(1.427)+2(1.507)+4(2)+2(4.1)+4(10.545)+28.44][/tex]
which returns an answer of:
L=32.32
Which is the length of the given curve.
