Determine the integral that can be used to find the arc length of the curve defined by the parametric equations [tex]x(t)=-4 \sqrt{3 t}[/tex] and [tex]y(t)=4 \sqrt{2 t+4}[/tex] from [tex]1 \leq t \leq 2[/tex].
Answer (2)
Find the derivatives of the parametric equations: d t d x = − 3 t 6 and d t d y = 2 t + 4 4 .
Square the derivatives: ( d t d x ) 2 = t 12 and ( d t d y ) 2 = t + 2 8 .
Substitute into the arc length formula: s = ∫ 1 2 t 12 + t + 2 8 d t .
The arc length integral is: s = ∫ 1 2 t 12 + t + 2 8 d t , so the answer is s = ∫ 1 2 t 12 + t + 2 8 d t .
Explanation
Problem Setup We are given the parametric equations x ( t ) = − 4 3 t and y ( t ) = 4 2 t + 4 , and we want to find the arc length of the curve from 1 ≤ t ≤ 2 . The formula for the arc length of a parametric curve is given by s = ∫ a b ( d t d x ) 2 + ( d t d y ) 2 d t where a and b are the limits of integration. In this case, a = 1 and b = 2 .
Finding Derivatives First, we need to find the derivatives of x ( t ) and y ( t ) with respect to t .
d t d x = d t d ( − 4 3 t ) = − 4 ⋅ 2 1 ( 3 t ) − 1/2 ⋅ 3 = − 6 ( 3 t ) − 1/2 = − 3 t 6 d t d y = d t d ( 4 2 t + 4 ) = 4 ⋅ 2 1 ( 2 t + 4 ) − 1/2 ⋅ 2 = 4 ( 2 t + 4 ) − 1/2 = 2 t + 4 4
Squaring Derivatives Next, we need to square these derivatives: ( d t d x ) 2 = ( − 3 t 6 ) 2 = 3 t 36 = t 12 ( d t d y ) 2 = ( 2 t + 4 4 ) 2 = 2 t + 4 16 = t + 2 8
Setting up the Arc Length Integral Now, we plug these into the arc length formula: s = ∫ 1 2 t 12 + t + 2 8 d t Thus, the integral that can be used to find the arc length is s = ∫ 1 2 t 12 + t + 2 8 d t
Final Answer Therefore, the integral to find the arc length is: s = ∫ 1 2 t 12 + t + 2 8 d t
Examples
Imagine you are designing a roller coaster. The track's length is crucial for determining the ride's duration and speed. By using parametric equations to define the track's curves and applying the arc length integral, you can accurately calculate the total length of the track. This ensures the roller coaster provides the desired thrill and safety for its riders.
To find the arc length of the curve defined by the given parametric equations from t = 1 to t = 2 , we use the arc length formula. The integral that can be set up for this is s = ∫ 1 2 t 12 + t + 2 8 d t . This integral represents the length of the curve over the specified interval.
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