Respuesta :
we are given
[tex]R(t)=cos(8t)i+sin(8t)j+8ln(cos(t))k[/tex]
now, we can find x , y and z components
[tex]x=cos(8t),y=sin(8t),z=8ln(cos(t))[/tex]
Arc length calculation:
we can use formula
[tex]L=\int\limits^a_b {\sqrt{(x')^2+(y')^2+(z')^2} } \, dt[/tex]
[tex]x'=-8sin(8t),y=8cos(8t),z=-8tan(t)[/tex]
now, we can plug these values
[tex]L=\int _0^{\frac{\pi }{4}}\sqrt{(-8sin(8t))^2+(8cos(8t))^2+(-8tan(t))^2} dt[/tex]
now, we can simplify it
[tex]L=\int _0^{\frac{\pi }{4}}\sqrt{64+64tan^2(t)} dt[/tex]
[tex]L=\int _0^{\frac{\pi }{4}}8\sqrt{1+tan^2(t)} dt[/tex]
[tex]L=\int _0^{\frac{\pi }{4}}8\sqrt{sec^2(t)} dt[/tex]
[tex]L=\int _0^{\frac{\pi }{4}}8sec(t) dt[/tex]
now, we can solve integral
[tex]\int \:8\sec \left(t\right)dt[/tex]
[tex]=8\ln \left|\tan \left(t\right)+\sec \left(t\right)\right|[/tex]
now, we can plug bounds
and we get
[tex]=8\ln \left(\sqrt{2}+1\right)-0[/tex]
so,
[tex]L=8\ln \left(1+\sqrt{2}\right)[/tex]..............Answer
The length of the curve is [tex]8ln(1+\sqrt{2})[/tex].
Given
The given curve is;
[tex]\rm \text{R(t) = cos(8t) i + sin(8t) j + 8 ln cost;} k, 0 \leq t\leq \dfrac{\pi }{4}[/tex]
What is differentiation?
The action or process of differentiating or distinguishing between two or more variables.
Now, differentiating the equation with respect to t;
[tex]\rm \text{R(t) = cos(8t) i + sin(8t) j + 8 ln cost} \\\\R'(t)=-8sin(8t)+8cos(8t)+(-8tant)\\\\[/tex]
Therefore,
The length of the curve is;
[tex]\rm R'(t)=-\sqrt{(8sin(8t))^2+(8cos(8t))^2+((-8tant))^2}\\\\ R'(t)=\sqrt{64sin^2(8t)+64cos^2(8t)+(64tan^2t)}}\\\\ R'(t)=\sqrt{64(sin^2(8t)+cos^2(8t))+(64tan^2t)}}\\\\ R'(t)=\sqrt{64(1)+(64tan^2t)}}\\\\ R'(t)=\sqrt{64+(64tan^2t)}\\\\ R'(t)=\sqrt{64(1+tan^2t)}\\\\R'(t)=8 \sqrt{sec^2t}\\\\R'(t)=8sect\\\\\text{Integrating the function both sides}\\\\\int {R'(t)} \ dt=\int {8sect} \ dt\\\\\rm R(t)=8 ln|tan(t)+sect|\\\\ substitute \pi/4 \ in \ the\ equation\\\\\rm R(t)=8ln(1+\sqrt{2})[/tex]
Hence, the length of the curve is [tex]8ln(1+\sqrt{2})[/tex].
To know more about Differentiation click the link given below.
https://brainly.com/question/17042788
