Find the arc length of the curve defined by the parametric equations [tex]x(t)=-2 e^{-2 t}[/tex] and [tex]y(t)=3 e^{-3 t}[/tex] from [tex]0.6 \leq t \leq 1.2[/tex]. Round your answer to the nearest hundredth.
Answer (2)
Find the derivatives: d t d x = 4 e − 2 t and d t d y = − 9 e − 3 t .
Substitute the derivatives into the arc length formula: L = ∫ 0.6 1.2 ( 4 e − 2 t ) 2 + ( − 9 e − 3 t ) 2 d t .
Simplify the expression inside the integral: L = ∫ 0.6 1.2 16 e − 4 t + 81 e − 6 t d t .
Evaluate the definite integral using numerical methods and round to the nearest hundredth: L ≈ 0.59 .
Explanation
Problem Setup We are given the parametric equations x ( t ) = − 2 e − 2 t and y ( t ) = 3 e − 3 t , and we want to find the arc length of the curve from t = 0.6 to t = 1.2 . The formula for the arc length L of a parametric curve is given by L = ∫ a b ( d t d x ) 2 + ( d t d y ) 2 d t where a and b are the limits of integration.
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 ( − 2 e − 2 t ) = − 2 ( − 2 ) e − 2 t = 4 e − 2 t d t d y = d t d ( 3 e − 3 t ) = 3 ( − 3 ) e − 3 t = − 9 e − 3 t
Substituting into Arc Length Formula Now, we substitute these derivatives into the arc length formula: L = ∫ 0.6 1.2 ( 4 e − 2 t ) 2 + ( − 9 e − 3 t ) 2 d t = ∫ 0.6 1.2 16 e − 4 t + 81 e − 6 t d t
Evaluating the Integral We need to evaluate the definite integral L = ∫ 0.6 1.2 16 e − 4 t + 81 e − 6 t d t This integral is difficult to solve analytically, so we will use numerical methods to approximate the value of the integral. Using a numerical integration method (Simpson's rule), we find that L ≈ 0.59
Final Answer Rounding the result to the nearest hundredth, we get L ≈ 0.59 .
Examples
Imagine you are designing a winding road for a scenic route. You have defined the road's path using parametric equations to ensure a smooth and visually appealing drive. To estimate the cost of paving the road, you need to calculate its length. By applying the arc length formula to the parametric equations, you can accurately determine the road's length, allowing for precise material estimations and cost planning. This ensures that you have enough resources to complete the paving project without overspending or running out of materials.
To find the arc length of the curve defined by the parametric equations, we calculated the derivatives and applied the arc length formula. After simplifying the integral, we used numerical methods to approximate the arc length. The final result, rounded to the nearest hundredth, is approximately 0.59.
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