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Chapter 14: Problem 32

Use a scalar line integral to find the length of the following curves. $$\mathbf{r}(t)=\langle 30 \sin t, 40 \sin t, 50 \cos t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Short Answer

Expert verified
Question: Determine the approximate length of the curve defined by the vector function $$\mathbf{r}(t)=\langle 30 \sin t, 40 \sin t, 50 \cos t\rangle$$ for t in the interval $$[0, 2\pi]$$. Answer: The approximate length of the curve can be found by evaluating the integral $$L = 100\int_{0}^{2\pi} \sqrt{9\cos^2 t + 16\cos^2 t + 25\sin^2 t}\, dt$$ using numerical integration techniques, such as Simpson's rule or a definite integral calculator software.

Step by step solution

01

Find the derivative of the vector function

To find the derivative of the vector function $$\mathbf{r}(t)=\langle 30 \sin t, 40 \sin t, 50 \cos t\rangle$$, we take the derivative of each component with respect to $$t$$: $$\frac{d\mathbf{r}}{dt} = \left\langle \frac{d(30\sin t)}{dt}, \frac{d(40\sin t)}{dt}, \frac{d(50\cos t)}{dt}\right\rangle$$ Now compute the derivatives for each component: $$\frac{d\mathbf{r}}{dt} = \langle 30\cos t, 40\cos t, -50\sin t\rangle$$
02

Calculate the magnitude of the derivative

Calculate the magnitude of the derivative vector $$\frac{d\mathbf{r}}{dt} = \langle 30\cos t, 40\cos t, -50\sin t\rangle$$ by using the formula for the magnitude of a vector: $$\left\lVert\frac{d\mathbf{r}}{dt}\right\rVert = \sqrt{(30\cos t)^2 + (40\cos t)^2 + (-50\sin t)^2}$$ $$\left\lVert\frac{d\mathbf{r}}{dt}\right\rVert = \sqrt{900\cos^2 t + 1600\cos^2 t + 2500\sin^2 t}$$
03

Set up the scalar line integral

With the magnitude of the derivative calculated, we can set up the scalar line integral to find the length of the curve. We need to integrate the magnitude of the derivative over the interval $$[0, 2\pi]$$: $$L = \int_{0}^{2\pi} \sqrt{900\cos^2 t + 1600\cos^2 t + 2500\sin^2 t}\, dt$$
04

Evaluate the integral

To find the length of the curve, we need to solve the integral: $$L = \int_{0}^{2\pi} \sqrt{900\cos^2 t + 1600\cos^2 t + 2500\sin^2 t}\, dt$$ We can factor out a 100 to simplify the expression inside the square root: $$L = 100\int_{0}^{2\pi} \sqrt{9\cos^2 t + 16\cos^2 t + 25\sin^2 t}\, dt$$ Since we have a square root and trigonometric functions inside the integral, this integral is quite challenging to solve analytically. However, we might resort to numerical methods to approximate the length of the curve. Using a numerical integration technique called Simpson's rule or any software that estimates definite integrals, we can approximate the length, L, of the curve.

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Most popular questions from this chapter

a. For what values of \(a, b, c,\) and \(d\) is the field \(\mathbf{F}=\langle a x+b y, c x+d y\rangle\) conservative? b. For what values of \(a, b,\) and \(c\) is the field \(\mathbf{F}=\left\langle a x^{2}-b y^{2}, c x y\right\rangle\) conservative?

Find a vector field \(\mathbf{F}\) with the given curl. In each case, is the vector field you found unique? $$\operatorname{curl} \mathbf{F}=\langle 0, z,-y\rangle$$

Prove that for a real number \(p,\) with \(\mathbf{r}=\langle x, y, z\rangle, \nabla\left(\frac{1}{|\mathbf{r}|^{p}}\right)=\frac{-p \mathbf{r}}{|\mathbf{r}|^{p+2}}.\)

For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. $$\mathbf{v}=\langle 0,-z, y\rangle$$

For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. $$v=\langle 0,0, y\rangle$$
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