r ( t ) = ( e t cos t , e t sin t , e t ) , for 0 ≤ t ≤ ln ( 4 ) r ( t ) is a parametric equation of x , y and z . r ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) ⇒ x ( t ) = e t cos t , y ( t ) = e t sin t , z ( t ) = e 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 ) = e t cos t , y ( t ) = e t sin t , z ( t ) = e t d x d t = e t ( cos t − sin t ) , d y d t = e t ( cos t + sin t ) , d z d t = e t s = ∫ 0 ln ( 4 ) e 2 t ( cos t − sin t ) 2 + e 2 t ( cos t − sin t ) 2 + e 2 t d t e 2 t ( cos t − sin t ) 2 + e 2 t ( cos t − sin t ) 2 + e 2 t = e 2 t ( ( cos t − sin t ) 2 + ( cos t − sin t ) 2 + 1 ) = e 2 t ( cos 2 t + sin 2 t − 2 cos t sin t + c o s 2 t + sin 2 t + 2 cos t sin t + 1 ) = e 2 t ( 1 + 1 + 1 ) = 3 e 2 t ∴ s = ∫ 0 ln ( 4 ) e 2 t ( 1 + 1 + 1 ) d t s = ∫ 0 ln ( 4 ) 3 e 2 t d t s = ∫ 0 ln ( 4 ) 3 e t d t s = 3 e t ∣ 0 ln ( 4 ) = 3 ( e ln ( 4 ) − e ( 0 ) ) = 3 ( 4 − 1 ) = 3 3 ∴ The length of the curve r ( t ) = ( e t cos t , e t sin t , e t ) , for 0 ≤ t ≤ ln ( 4 ) is 3 3 units ≈ 5.1962 units r(t) = (\displaystyle e^t\cos{t}, e^t\sin{t}, e^t), \hspace{0.2cm} \textsf{for}\hspace{0.1cm} 0\leq t \leq\ln(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. \\
r(t) = (x(t), y(t), z(t)) \Rightarrow x(t) = e^t\cos{t}, y(t)=e^t\sin{t}, z(t)=e^t\\
\textsf{Length of the curve \textit{(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) = e^t\cos{t}, y(t)=e^t\sin{t}, z(t)=e^t\\
\frac{\mathrm{d}x}{\mathrm{d}t} = e^t(\cos{t} -\sin{t}), \frac{\mathrm{d}y}{\mathrm{d}t} =e^t(\cos{t} + \sin{t}), \frac{\mathrm{d}z}{\mathrm{d}t} = e^t\\
s = \int_{0}^{\ln(4)}\sqrt{e^{2t}(\cos{t} -\sin{t})^2 + e^{2t}(\cos{t} -\sin{t})^2+ e^{2t}}\hspace{0.1cm} \mathrm{d}t\\
e^{2t}(\cos{t} -\sin{t})^2 + e^{2t}(\cos{t} -\sin{t})^2+ e^{2t} \\= e^{2t}((\cos{t} -\sin{t})^2 + (\cos{t} -\sin{t})^2+ 1) \\= e^{2t}(\cos^2{t} + \sin^2{t} - 2\cos{t}\sin{t} + cos^2{t} + \sin^2{t} + 2\cos{t}\sin{t} + 1)\\ = e^{2t}(1 + 1 + 1) = 3e^{2t}\\
\therefore s = \int_{0}^{\ln(4)} \sqrt{e^{2t}(1 + 1 + 1)}\hspace{0.1cm} \mathrm{d}t\\
s = \int_{0}^{\ln(4)} \sqrt{3}\sqrt{e^{2t}} \hspace{0.1cm} \mathrm{d}t\\
s = \int_{0}^{\ln(4)} \sqrt{3} e^t \hspace{0.1cm} \mathrm{d}t\\
s = \sqrt{3}e^t \vert_{0}^{\ln(4)} = \sqrt{3}(e^{\ln(4)} - e^{(0)})= \sqrt{3}(4 - 1) = 3\sqrt{3}\\
\therefore \textsf{The length of the curve}\hspace{0.1cm} r(t) = (e^t\cos{t}, e^t\sin{t}, e^t), \hspace{0.2cm} \textsf{for}\hspace{0.1cm} 0\leq t \leq \ln(4)\hspace{0.1cm}\\\textsf{is}\hspace{0.1cm} 3\sqrt{3}\hspace{0.1cm}\textsf{units} \approx 5.1962 \hspace{0.1cm}\textsf{units} r ( t ) = ( e t cos t , e t sin t , e t ) , for 0 ≤ t ≤ ln ( 4 ) r ( t ) is a parametric equation of x , y and z . r ( t ) = ( x ( t ) , y ( t ) , z ( t )) ⇒ x ( t ) = e t cos t , y ( t ) = e t sin t , z ( t ) = e 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 ) = e t cos t , y ( t ) = e t sin t , z ( t ) = e t d t d x = e t ( cos t − sin t ) , d t d y = e t ( cos t + sin t ) , d t d z = e t s = ∫ 0 l n ( 4 ) e 2 t ( cos t − sin t ) 2 + e 2 t ( cos t − sin t ) 2 + e 2 t d t e 2 t ( cos t − sin t ) 2 + e 2 t ( cos t − sin t ) 2 + e 2 t = e 2 t (( cos t − sin t ) 2 + ( cos t − sin t ) 2 + 1 ) = e 2 t ( cos 2 t + sin 2 t − 2 cos t sin t + co s 2 t + sin 2 t + 2 cos t sin t + 1 ) = e 2 t ( 1 + 1 + 1 ) = 3 e 2 t ∴ s = ∫ 0 l n ( 4 ) e 2 t ( 1 + 1 + 1 ) d t s = ∫ 0 l n ( 4 ) 3 e 2 t d t s = ∫ 0 l n ( 4 ) 3 e t d t s = 3 e t ∣ 0 l n ( 4 ) = 3 ( e l n ( 4 ) − e ( 0 ) ) = 3 ( 4 − 1 ) = 3 3 ∴ The length of the curve r ( t ) = ( e t cos t , e t sin t , e t ) , for 0 ≤ t ≤ ln ( 4 ) is 3 3 units ≈ 5.1962 units
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