Answer to Question #97450 in Differential Geometry | Topology for Warsi

Question #97450
find the arc length of r(t)=(e t sin t , e t cos t ,e t)
1
Expert's answer
2019-11-08T06:19:18-0500

"x(t)=e^t\\sin(t)"

"y(t)=e^t\\cos(t)"

"z(t)=e^t"


"x'(t)=e^t\\sin(t)+e^t\\cos(t)"

"y'(t)=e^t\\cos(t)-e^t\\sin(t)"

"z'(t)=e^t"


arc length of "r(t) = \\intop _a^b\\sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2}dt="

"=\\intop _a^b\\sqrt{(e^t\\sin{t}+e^t\\cos{t})^2+(e^t\\cos{t}-e^t\\sin{t})^2+(e^t)^2}dt="

"=\\intop _a^b\\sqrt{e^{2t}\\sin^{2}{t}+2e^{2t}\\sin{t}\\cos{t}+e^{2t}\\cos^{2}{t}+e^{2t}\\cos^{2}{t}-2e^{2t}\\sin{t}\\cos{t}+e^{2t}\\sin^{2}{t}+e^{2t}}dt="

"=\\intop _a^b\\sqrt{2e^{2t}\\sin^{2}{t}+2e^{2t}\\cos^{2}{t}+e^{2t}}dt="

"=\\intop _a^b\\sqrt{2e^{2t}(\\sin^{2}{t}+\\cos^{2}{t})+e^{2t}}dt="

"=\\intop _a^b\\sqrt{2e^{2t}+e^{2t}}dt="

"=\\intop _a^b\\sqrt{3e^{2t}}dt="

"=\\sqrt{3}\\intop _a^b{e^{t}}dt="

"=\\sqrt{3}{e^{t}}|\\ _a^b="

"=\\sqrt{3}(e^{b}-e^{a})"


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