Find the arc length for $\mathbf { r } ( t ) = \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 8 t ^ { 3 } \mathbf { k } \text { over the interval } [ 0,2 ] \text {. Round your }$ answer to two decimal places.
A)67.63
B)73.37
C)79.40
D)49.75
E)62.97

Question

Quizplus · Accepted Answer

Explanation:The arc length formula is given by$$s = \int_a^b \Vert \mathbf { r } ' ( t ) \Vert dt$$So, for the given vector function, we have$$\mathbf { r } ' ( t ) = 0 \mathbf { i } + 10 t \mathbf { j } + 24 t ^ { 2 } \mathbf { k }$$and$$\Vert \mathbf { r } ' ( t ) \Vert = \sqrt { 0 ^ { 2 } + ( 10 t ) ^ { 2 } + ( 24 t ^ { 2 } ) ^ { 2 } } = \sqrt { 576 t ^ { 4 } + 100 t ^ { 2 } }$$Therefore, the arc length is$$s = \int_0^2 \sqrt { 576 t ^ { 4 } + 100 t ^ { 2 } } dt$$This integral cannot be evaluated in terms of elementary functions, so we must use numerical integration. Using a calculator, we find that$$s \approx 67.63$$Therefore, the correct choice is A.

Find the Arc Length For r(t)=i+5t2j+8t3k over the interval [0,2]. Round your \mathbf { r } ( t ) = \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 8 t ^ { 3 } \mathbf { k } \text { over the interval } [ 0,2 ] \text {. Round your }r(t)=i+5t2j+8t3k over the interval [0,2]. Round your

Find the Arc Length For $\mathbf { r } ( t ) = \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 8 t ^ { 3 } \mathbf { k } \text { over the interval } [ 0,2 ] \text {. Round your }$