x ( t ) = e t sin ( t ) x(t)=e^t\sin(t) x ( t ) = e t sin ( t )
y ( t ) = e t cos ( t ) y(t)=e^t\cos(t) y ( t ) = e t cos ( t )
z ( t ) = e t z(t)=e^t z ( t ) = e t
x ′ ( t ) = e t sin ( t ) + e t cos ( t ) x'(t)=e^t\sin(t)+e^t\cos(t) x ′ ( t ) = e t sin ( t ) + e t cos ( t )
y ′ ( t ) = e t cos ( t ) − e t sin ( t ) y'(t)=e^t\cos(t)-e^t\sin(t) y ′ ( t ) = e t cos ( t ) − e t sin ( t )
z ′ ( t ) = e t z'(t)=e^t z ′ ( t ) = e t
arc length of r ( t ) = ∫ a b ( x ′ ( t ) ) 2 + ( y ′ ( t ) ) 2 + ( z ′ ( t ) ) 2 d t = r(t) = \intop _a^b\sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2}dt= r ( t ) = ∫ a b ( x ′ ( t ) ) 2 + ( y ′ ( t ) ) 2 + ( z ′ ( t ) ) 2 d t =
= ∫ a b ( e t sin t + e t cos t ) 2 + ( e t cos t − e t sin t ) 2 + ( e t ) 2 d t = =\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= = ∫ a b ( e t sin t + e t cos t ) 2 + ( e t cos t − e t sin t ) 2 + ( e t ) 2 d t =
= ∫ a b e 2 t sin 2 t + 2 e 2 t sin t cos t + e 2 t cos 2 t + e 2 t cos 2 t − 2 e 2 t sin t cos t + e 2 t sin 2 t + e 2 t d t = =\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= = ∫ a b e 2 t sin 2 t + 2 e 2 t sin t cos t + e 2 t cos 2 t + e 2 t cos 2 t − 2 e 2 t sin t cos t + e 2 t sin 2 t + e 2 t d t =
= ∫ a b 2 e 2 t sin 2 t + 2 e 2 t cos 2 t + e 2 t d t = =\intop _a^b\sqrt{2e^{2t}\sin^{2}{t}+2e^{2t}\cos^{2}{t}+e^{2t}}dt= = ∫ a b 2 e 2 t sin 2 t + 2 e 2 t cos 2 t + e 2 t d t =
= ∫ a b 2 e 2 t ( sin 2 t + cos 2 t ) + e 2 t d t = =\intop _a^b\sqrt{2e^{2t}(\sin^{2}{t}+\cos^{2}{t})+e^{2t}}dt= = ∫ a b 2 e 2 t ( sin 2 t + cos 2 t ) + e 2 t d t =
= ∫ a b 2 e 2 t + e 2 t d t = =\intop _a^b\sqrt{2e^{2t}+e^{2t}}dt= = ∫ a b 2 e 2 t + e 2 t d t =
= ∫ a b 3 e 2 t d t = =\intop _a^b\sqrt{3e^{2t}}dt= = ∫ a b 3 e 2 t d t =
= 3 ∫ a b e t d t = =\sqrt{3}\intop _a^b{e^{t}}dt= = 3 ∫ a b e t d t =
= 3 e t ∣ a b = =\sqrt{3}{e^{t}}|\ _a^b= = 3 e t ∣ a b =
= 3 ( e b − e a ) =\sqrt{3}(e^{b}-e^{a}) = 3 ( e b − e a )
#339914 on Dec 2023
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