Question

Explain how to find the length of the curve \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle,\) for \(a \leq t \leq b\)

Short answer

Answer: The main concept used to compute the length of the curve described by a vector function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) between the points \(t=a\) and \(t=b\) is the framework of calculus to compute the length of the curve through integrals.

Step-by-step solution

Compute the derivative of the vector function

First, we need to find the derivative of the vector function \(\mathbf{r}(t)\) with respect to \(t\). To do so, simply differentiate each component of the vector function with respect to \(t\): $$ \frac{d\mathbf{r}}{dt} = \left\langle \frac{df}{dt}, \frac{dg}{dt}, \frac{dh}{dt} \right\rangle $$

Find the magnitude of the derivative

We need to find the size of the vector generated in Step 1. This is known as the magnitude, which is typically denoted by \(||\mathbf{v}||\). To compute the magnitude of the derivative vector, use the following formula: $$ ||\frac{d\mathbf{r}}{dt}|| = \sqrt{\left(\frac{df}{dt}\right)^2 + \left(\frac{dg}{dt}\right)^2 + \left(\frac{dh}{dt}\right)^2} $$

Integrate the magnitude of the derivative

The final step is to integrate the magnitude of the derivative obtained in Step 2. This will give us the length of the curve, which can be represented as follows: $$ L = \int_{a}^{b} ||\frac{d\mathbf{r}}{dt}|| dt = \int_{a}^{b} \sqrt{\left(\frac{df}{dt}\right)^2 + \left(\frac{dg}{dt}\right)^2 + \left(\frac{dh}{dt}\right)^2} dt $$ The integral can sometimes be solved analytically using known integration techniques, or it might require numerical methods if the integration is too complex. Regardless, once the integral is evaluated, you will have the length of the curve between the points \(t=a\) and \(t=b\).

Key concepts

Curve Length

Finding the length of a curve is an important concept in vector calculus. When dealing with curves defined by vector functions, such as \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\), the length of the curve from \(t=a\) to \(t=b\) can be found using calculus. The process begins with calculating the derivative of the vector function.

Curve length requires the integration of the magnitude of this derivative. The integral \[L = \int_{a}^{b} \sqrt{\left(\frac{df}{dt}\right)^2 + \left(\frac{dg}{dt}\right)^2 + \left(\frac{dh}{dt}\right)^2} dt\]provides the total length by accumulating infinitesimal distances along the curve. Each small segment of the curve acts like a tiny straight line, and the integral sums these segments to give a total distance.

Vector Derivatives

Taking the derivative of a vector function involves differentiating each component separately. If you have a vector \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\), its derivative \(\frac{d\mathbf{r}}{dt}\) is computed component-wise:

\[\frac{d\mathbf{r}}{dt} = \left\langle \frac{df}{dt}, \frac{dg}{dt}, \frac{dh}{dt} \right\rangle\]This process gives you another vector that describes the direction and rate of change of the original vector function at each point.

Understanding vector derivatives is crucial because they are used to calculate the magnitude of the derivative vector, an essential step in finding the length of a curve, as well as in other applications like velocity and acceleration.

Integration Techniques

Integration is the mathematical method used to find accumulated values, such as areas under curves or, in our case, the total length of a curve. When calculating curve length, the integral involves the magnitude of the vector derivative:

\[\int_{a}^{b} \sqrt{\left(\frac{df}{dt}\right)^2 + \left(\frac{dg}{dt}\right)^2 + \left(\frac{dh}{dt}\right)^2} dt\]Evaluating this integral can be done using analytical techniques if the function is simple enough, but more complex integrals might require numerical methods like numerical integration or approximation techniques.

Numerical methods can involve Riemann sums, trapezoidal rule, or Simpson's rule, which approximate the area under a curve when an exact integral is hard to find. These methods increase in accuracy with the number of partitions used during approximation.