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Vector Valued Functions Calculator

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Example
Created on 2024-06-20Asked by Mila Sanchez (Solvelet student)
Find the arc length of the vector function r(t)=⟨cos⁡(t),sin⁡(t),t⟩\mathbf{r}(t) = \langle \cos(t), \sin(t), t \rangle for 0≤t≤2π0 \leq t \leq 2\pi.

Solution

To find the arc length of the vector function r(t)=⟨cos⁡(t),sin⁡(t),t⟩\mathbf{r}(t) = \langle \cos(t), \sin(t), t \rangle for 0≤t≤2π0 \leq t \leq 2\pi: 1. **Find the derivative of r(t)\mathbf{r}(t):** r′(t)=⟨−sin⁡(t),cos⁡(t),1⟩. \mathbf{r}'(t) = \left\langle -\sin(t), \cos(t), 1 \right\rangle. 2. **Compute the magnitude of r′(t)\mathbf{r}'(t):** ∥r′(t)∥=(−sin⁡(t))2+(cos⁡(t))2+12=sin⁡2(t)+cos⁡2(t)+1=1+1=2. \|\mathbf{r}'(t)\| = \sqrt{(-\sin(t))^2 + (\cos(t))^2 + 1^2} = \sqrt{\sin^2(t) + \cos^2(t) + 1} = \sqrt{1 + 1} = \sqrt{2}. 3. **Integrate the magnitude over the given interval:** L=∫02π∥r′(t)∥ dt=∫02π2 dt=2∫02πdt=2⋅2π=2π2. L = \int_{0}^{2\pi} \|\mathbf{r}'(t)\| \, dt = \int_{0}^{2\pi} \sqrt{2} \, dt = \sqrt{2} \int_{0}^{2\pi} dt = \sqrt{2} \cdot 2\pi = 2\pi\sqrt{2}. 4. **Result:** The arc length of the vector function r(t)=⟨cos⁡(t),sin⁡(t),t⟩\mathbf{r}(t) = \langle \cos(t), \sin(t), t \rangle for 0≤t≤2π0 \leq t \leq 2\pi is 2π22\pi\sqrt{2}. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Noah Gonzalez on Solvelet
1. Find the velocity and acceleration vectors for the curve described by the vector-valued function r(t)=⟨cos⁡(t),sin⁡(t),t⟩\mathbf{r}(t)=\langle \cos(t), \sin(t), t \rangle.2. Determine the unit tangent vector for the curve described by the vector-valued function r(t)=⟨et,cos⁡(t),sin⁡(t)⟩ \mathbf{r}(t) = \langle e^t, \cos(t), \sin(t) \rangle .,
DefinitionVector-valued function -- a function that spits out a vector for every point in a domain, and when we are modelling things in physical space maybe we want to model it because physical space corresponds to vectors (and more vectors). Example : r(t) = is a parametric form of helix Get advanced SolveletAI step-by-step solutions with Applied Vector Valued Functions (Berkeley Math 1A).DialogInterface.
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