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Arc Length of Vector Functions Calculator

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Example
Created on 2024-06-20Asked by Sofia Martin (Solvelet student)
Find the arc length of the vector function r(t)=⟨3t,4sin⁡(t),4cos⁡(t)⟩ \mathbf{r}(t) = \langle 3t, 4\sin(t), 4\cos(t) \rangle for 0≤t≤π 0 \le t \le \pi .

Solution

Step 1: Identify the vector function and the interval. The vector function is r(t)=⟨3t,4sin⁡(t),4cos⁡(t)⟩ \mathbf{r}(t) = \langle 3t, 4\sin(t), 4\cos(t) \rangle and the interval is 0≤t≤π 0 \le t \le \pi . Step 2: Find the derivative of the vector function: r′(t)=⟨3,4cos⁡(t),−4sin⁡(t)⟩. \mathbf{r}'(t) = \langle 3, 4\cos(t), -4\sin(t) \rangle. Step 3: Find the magnitude of the derivative: ∥r′(t)∥=32+(4cos⁡(t))2+(−4sin⁡(t))2=9+16cos⁡2(t)+16sin⁡2(t)=9+16=25=5. \left\| \mathbf{r}'(t) \right\| = \sqrt{3^2 + (4\cos(t))^2 + (-4\sin(t))^2} = \sqrt{9 + 16\cos^2(t) + 16\sin^2(t)} = \sqrt{9 + 16} = \sqrt{25} = 5. Step 4: Set up the arc length integral: ∫0π∥r′(t)∥ dt=∫0π5 dt. \int_{0}^{\pi} \left\| \mathbf{r}'(t) \right\| \, dt = \int_{0}^{\pi} 5 \, dt. Step 5: Evaluate the integral: 5∫0π dt=5t∣0π=5π−0=5π. 5 \int_{0}^{\pi} \, dt = 5t \bigg|_{0}^{\pi} = 5\pi - 0 = 5\pi. Step 6: Conclusion. The arc length of the vector function r(t)=⟨3t,4sin⁡(t),4cos⁡(t)⟩ \mathbf{r}(t) = \langle 3t, 4\sin(t), 4\cos(t) \rangle for 0≤t≤π 0 \le t \le \pi is 5π 5\pi . Solved on Solvelet with Basic AI Model
Some of the related questions asked by Chloe Hernandez on Solvelet
1. Determine the arc length of the vector function r(t)=⟨t,t2,t3⟩ \mathbf{r}(t) = \langle t, t^2, t^3 \rangle over the interval [0,1] [0, 1] 2. Calculate the arc length of the trajectory of a particle moving along the curve defined by the vector function r(t)=⟨cos⁡(t),sin⁡(t),t⟩ \mathbf{r}(t) = \langle \cos(t), \sin(t), t \rangle over the interval [0,π] [0, \pi] .,
DefinitionThe arc length s of a vector function r(t) is the length of the curve traced out by the vector as t varies over a specified interval. The arc length of a vector function is FFF defined as the FFF integral of its derivative magnitude with respect to the parameter ttt; it is used in physics to describe how particles move, in engineering to calculate trajectory shapes, and in computer graphics in path animations. For example, in projectile motion, we find the arc length of the trajectory of a projectile using the parametrics that describe its motion
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