To determine Find the length of the given smooth curve. Explanation Given information: The smooth curve is, r ( t ) = ( 3 cos t ) i + ( 3 sin t ) j + 2 t 3 2 k , 0 ≤ t ≤ 3 Formula used: Write the expression to calculate the velocity vector . v = d r d t (1) Write the expression to calculate the arc length of the curve. L = ∫ a b | v | d t (2) Calculation: Substitute the given curve function in equation (1) to find v . v = d d t ( ( 3 cos t ) i + ( 3 sin t ) j + 2 t 3 2 k ) = 3 ( − sin t ) i + 3 ( cos t ) j + 2 ( 3 2 t 3 2 − 1 ) k = ( − 3 sin t ) i + ( 3 cos t ) j + 3 t 3 − 2 2 k = ( − 3 sin t ) i + ( 3 cos t ) j + 3 t 1 2 k To find | v | : | v | = | ( − 3 sin t ) i + ( 3 cos t ) j + 3 t 1 2 k | = ( − 3 sin t ) 2 + ( 3 cos t ) 2 + ( 3 t 1 2 ) 2 = 9 sin 2 t + 9 cos 2 t + 9 t 2 2 = 9 ( sin 2 t + cos 2 t ) + 9 t Simplify the above equation to find | v |

4th Edition

ISBN: 9780134439099

Author: Hass, Joel., Heil, Christopher , WEIR, Maurice D.

Publisher: Pearson,

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