- #1
Sho Kano
- 372
- 3
Homework Statement
Find the arc-length parameterization for [itex]r(t)=\left< { e }^{ 2t },{ e }^{ -2t },2\sqrt { 2 } t \right> ,t\ge 0[/itex]
Homework Equations
[itex]s(t)=\int { \left| \dot { r } (t) \right| dt } [/itex]
The Attempt at a Solution
[itex]\dot { r } (t)=\left< { 2e }^{ 2t },-2{ e }^{ -2t },2\sqrt { 2 } \right> \\ \left| \dot { r } (t) \right| =\sqrt { 4{ e }^{ 4t }+4{ e }^{ -4t }+8 } \\ =2\sqrt { { e }^{ 4t }+{ e }^{ -4t }+2 } \\ =2\sqrt { { e }^{ 4t }+\frac { 1 }{ { e }^{ 4t } } +2 } \\ =2\sqrt { \frac { { e }^{ 8t }+{ 2e }^{ 4t }+1 }{ { e }^{ 4t } } } =2\sqrt { \frac { \left( { e }^{ 4t }+1 \right) \left( { e }^{ 4t }+1 \right) }{ { e }^{ 4t } } } \\ =\frac { 2 }{ { e }^{ 2t } } \left( { e }^{ 4t }+1 \right) \\ ={ 2e }^{ 2t }+{ 2e }^{ -2t }\\ s(t)-s(0)=\int _{ 0 }^{ t }{ { 2e }^{ 2t }+{ 2e }^{ -2t }dt } =0[/itex] It can't equal 0...where did I go wrong?