r ( t ) = ( cos t , sin t , ln ( cos t ) ) , for 0 ≤ t ≤ π 4 r ( t ) is a parametric equation of x , y and z . Also note that r ( t ) : R → R 3 is a vector valued function of a real variable with independent scalar output variables x , y & z r ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) ⇒ x ( t ) = cos t , y ( t ) = sin t , z ( t ) = ln cos t Length of the curve ( s ) = ∫ s 1 s 2 ( d x d t ) 2 + ( d y d t ) 2 + ( d z d t ) 2 d t x ( t ) = cos t , y ( t ) = sin t , z ( t ) = ln ( cos t ) d x d t = − sin t , d y d t = cos t , d z d t = − tan t s = ∫ 0 π 4 sin 2 t + cos 2 t + tan 2 t d t s = ∫ 0 π 4 1 + tan 2 t d t s = ∫ 0 π 4 sec 2 t d t s = ∫ 0 π 4 sec t d t s = ∫ 0 π 4 sec t ( sec t + tan t sec t + tan t ) d t s = ∫ 0 π 4 sec 2 t + tan t sec t sec t + tan t d t s = ∫ 0 π 4 d ( sec t + tan t ) sec t + tan t s = ln ( sec t + tan t ) ∣ 0 π 4 s = ln ( sec ( π 4 ) + tan ( π 4 ) ) − ln ( sec 0 + tan 0 ) s = ln ( 2 + 1 ) − ln ( 1 + 0 ) = ln ( 2 + 1 ) − ln ( 1 ) = ln ( 2 + 1 ) − 0 = ln ( 2 + 1 ) ∴ The length of the curve r ( t ) = ( cos t , sin t , ln cos t ) ) , for 0 ≤ t ≤ π 4 is ln ( 2 + 1 ) unit ≈ 0.8814 unit r(t) = (\displaystyle\cos{t}, \sin{t}, \ln(\cos{t})), \hspace{0.2cm} \textsf{for}\hspace{0.1cm} 0\leq t \leq \frac{\pi}{4}\\
r(t)\hspace{0.1cm}\textsf{is a parametric equation of}\hspace{0.1cm} x, y\hspace{0.1cm} \textsf{and}\hspace{0.1cm} z. \\\textsf{Also note that}\hspace{0.1cm} r(t): \mathbb{R} \rightarrow \mathbb{R}^3 \hspace{0.1cm} \\\textsf{is a vector valued function of a real variable}\\
\textsf{with independent scalar output variables} \hspace{0.1cm} x, y \hspace{0.1cm}\&\hspace{0.1cm} z\\
r(t) = (x(t), y(t), z(t)) \Rightarrow x(t) = \cos{t}, y(t)=\sin{t}, z(t)=\ln{\cos{t}}\\
\textsf{Length of the curve}\hspace{0.1cm} (s) = \int_{s_1}^{s_2} \sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2 + \left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2 + \left(\frac{\mathrm{d}z}{\mathrm{d}t}\right)^2 }\hspace{0.1cm} \mathrm{d}t\\
x(t) = \cos{t}, y(t)=\sin{t}, z(t)=\ln(\cos{t}) \\
\frac{\mathrm{d}x}{\mathrm{d}t} = -\sin{t}, \frac{\mathrm{d}y}{\mathrm{d}t} = \cos{t}, \frac{\mathrm{d}z}{\mathrm{d}t} = -\tan{t}\\
s = \int_{0}^{\frac{\pi}{4}} \sqrt{\sin^2{t} + \cos^2{t} + \tan^2{t}}\hspace{0.1cm} \mathrm{d}t\\
s = \int_{0}^{\frac{\pi}{4}} \sqrt{1 + \tan^2{t}}\hspace{0.1cm} \mathrm{d}t\\
s = \int_{0}^{\frac{\pi}{4}} \sqrt{\sec^2{t}} \hspace{0.1cm} \mathrm{d}t\\
s = \int_{0}^{\frac{\pi}{4}} \sec{t} \hspace{0.1cm} \mathrm{d}t\\
s = \int_{0}^{\frac{\pi}{4}} \sec{t}\left(\frac{\sec{t} + \tan{t}}{\sec{t} + \tan{t}}\right) \mathrm{d}t\\
s = \int_{0}^{\frac{\pi}{4}} \frac{\sec^2{t} + \tan{t}\sec{t}}{\sec{t} + \tan{t}}\hspace{0.1cm} \mathrm{d}t\\
s = \int_{0}^{\frac{\pi}{4}} \frac{\mathrm{d}(\sec{t} + \tan{t})}{\sec{t} + \tan{t}}\\
s = \ln(\sec{t} + \tan{t})\vert_{0}^{\frac{\pi}{4}}\\
s = \ln\left(\sec\left(\frac{\pi}{4}\right) + \tan\left(\frac{\pi}{4}\right)\right) - \\\ln(\sec{0} + \tan{0})\\
s = \ln(\sqrt{2} + 1) - \ln(1 + 0) = \\\ln(\sqrt{2} + 1) - \ln(1) =\\ \ln(\sqrt{2} + 1) - 0 = \ln(\sqrt{2} + 1)\\
\therefore \textsf{The length of the curve}\hspace{0.1cm} r(t) = (\cos{t}, \sin{t}, \ln{\cos{t}})), \hspace{0.2cm} \textsf{for}\hspace{0.1cm} 0\leq t \leq \frac{\pi}{4}\hspace{0.1cm}\\\textsf{is}\hspace{0.1cm} \ln(\sqrt{2} + 1)\hspace{0.1cm}\textsf{unit} \approx 0.8814 \hspace{0.1cm}\textsf{unit} r ( t ) = ( cos t , sin t , ln ( cos t )) , for 0 ≤ t ≤ 4 π r ( t ) is a parametric equation of x , y and z . Also note that r ( t ) : R → R 3 is a vector valued function of a real variable with independent scalar output variables x , y & z r ( t ) = ( x ( t ) , y ( t ) , z ( t )) ⇒ x ( t ) = cos t , y ( t ) = sin t , z ( t ) = ln cos t Length of the curve ( s ) = ∫ s 1 s 2 ( d t d x ) 2 + ( d t d y ) 2 + ( d t d z ) 2 d t x ( t ) = cos t , y ( t ) = sin t , z ( t ) = ln ( cos t ) d t d x = − sin t , d t d y = cos t , d t d z = − tan t s = ∫ 0 4 π sin 2 t + cos 2 t + tan 2 t d t s = ∫ 0 4 π 1 + tan 2 t d t s = ∫ 0 4 π sec 2 t d t s = ∫ 0 4 π sec t d t s = ∫ 0 4 π sec t ( sec t + tan t sec t + tan t ) d t s = ∫ 0 4 π sec t + tan t sec 2 t + tan t sec t d t s = ∫ 0 4 π sec t + tan t d ( sec t + tan t ) s = ln ( sec t + tan t ) ∣ 0 4 π s = ln ( sec ( 4 π ) + tan ( 4 π ) ) − ln ( sec 0 + tan 0 ) s = ln ( 2 + 1 ) − ln ( 1 + 0 ) = ln ( 2 + 1 ) − ln ( 1 ) = ln ( 2 + 1 ) − 0 = ln ( 2 + 1 ) ∴ The length of the curve r ( t ) = ( cos t , sin t , ln cos t )) , for 0 ≤ t ≤ 4 π is ln ( 2 + 1 ) unit ≈ 0.8814 unit
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