Question

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

Short answer

Question: Find the length of the curve given by the parametric equations \(\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle\) for \(a \leq t \leq b\). Answer: To find the length of the curve, start by computing the derivative of the position vector \(\mathbf{r}(t)\) with respect to \(t\). Then, find the magnitude of the tangent vector at each point. Next, apply the arc length formula and evaluate the integral over the interval \([a, b]\). The result will be the length of the curve from \(a\) to \(b\).

Step-by-step solution

Compute the derivative of the position vector

Differentiate the position vector \(\mathbf{r}(t)\) with respect to \(t\). The derivative will give us the tangent vector at each point on the curve. $$ \frac{d\mathbf{r}}{dt} = \left\langle \frac{df(t)}{dt}, \frac{dg(t)}{dt}, \frac{dh(t)}{dt} \right\rangle $$

Find the magnitude of the tangent vector

In order to find the arc length of the curve, we need to know the magnitude of the tangent vector at each point. To do this, we take the square root of the sum of the squares of the components of the tangent vector. $$ \left\|\frac{d\mathbf{r}}{dt}\right\| = \sqrt{\left(\frac{df(t)}{dt}\right)^2 + \left(\frac{dg(t)}{dt}\right)^2 + \left(\frac{dh(t)}{dt}\right)^2} $$

Apply the arc length formula

Using the arc length formula, we can write the length of the curve in terms of an integral with respect to \(t\). Note that the bounds of the integral are given by the problem statement: \(a \leq t \leq b\). $$ L = \int_a^b \left\|\frac{d\mathbf{r}}{dt}\right\| \, dt = \int_a^b \sqrt{\left(\frac{df(t)}{dt}\right)^2 + \left(\frac{dg(t)}{dt}\right)^2 + \left(\frac{dh(t)}{dt}\right)^2} \, dt $$

Evaluate the integral

Depending on the functions \(f(t)\), \(g(t)\), and \(h(t)\) given, the integral can be evaluated using various techniques such as direct integration, substitution, integration by parts, etc. Once the integral is evaluated, the result will be the length of the curve from \(a\) to \(b\). In summary, finding the length of the curve involves finding the magnitude of the derivative of the position vector, and then using the arc length formula to set up an integral with respect to \(t\). Finally, evaluating that integral will give you the length of the curve over the interval \([a, b]\).

Key concepts

Tangent Vector

Visualizing the path of a moving object can be quite fascinating, especially when that object is tracing a curve through space. To thoroughly understand this path, we introduce the concept of a tangent vector.

Imagine you're on a train smoothly curving along a railway track; at any given moment, the direction the train is headed is analogous to the tangent vector at that point on the track. It's a vector that just skims the curve, pointing in the direction of motion at any instance. Mathematically, you obtain the tangent vector by differentiating the position vector, \textbf{r}(t), with respect to the parameter, t. The result of this derivative is a new vector that captures the instantaneous rate of change of the curve in each of its three dimensions, represented as \( \frac{d\textbf{r}}{dt} \).

Now consider how we computed the derivative in our exercise; this process is key to determining how sharply our curve bends and where it's headed next. Without this step, calculating the arc length, which is the distance along the curve, would be impossible. The tangent vector's direction is vital, but for arc length, knowing its size, or magnitude, is what truly matters, leading us comfortably into our next topic.

Magnitude of a Vector

Knowing the direction is crucial, but how far does our curve go? That's where the magnitude of a vector comes into play. Think of magnitude as the length of our tangent vector; it gives us information on how much change occurs at each point on our curve.

The magnitude is found by taking the square root of the sum of its components squared, as you noticed when we calculated \( \left\|\frac{d\textbf{r}}{dt}\right\| \). This mathematical expression resembles the Pythagorean theorem in three dimensions. It provides us with a scalar quantity: a single number that represents the vector's length in space. It's akin to pulling a taut string along the path your train takes and measuring its length; the magnitude tells us how long the string is at any point. We use this magnitude as a building block for determining the entire curve's length in the arc length formula, bridging the gap between a single point and the whole journey from point a to point b.

Derivative of a Vector Function

The derivative of a vector function holds the key to unlocking the curve's secrets. The process of differentiation, familiar from standard calculus, extends beautifully to vector functions. When we differentiate, we're not just looking for a rate of change in a single variable. Instead, we're discovering how each component of our vector—the functions f(t), g(t), and h(t)—evolves as time t marches on.

The derivative of our vector function \( \mathbf{r}(t) \) is the trio of these rates of changes, or slopes if you will, forming a new vector that embodies the curve's instantaneous direction and speed. In the context of our exercise, we're especially interested in how quickly the curve is moving away from its start point at any given time. This information is vital in piecing together the curve's length over its entire span, cementing the derivative's role as an invaluable tool in our mathematical toolkit.

Integral Calculus

So how do we go from understanding a curve's behavior at a point to comprehending its entire stretch? Enter integral calculus, the branch of mathematics dedicated to accumulation. If differentiation is about splitting things up to understand the instantaneous, integration is about putting them together to understand the whole.

The arc length formula is a prime example of integral calculus at work. By integrating the magnitude of the tangent vector, we accumulate the little bits of length along the curve to find its total length within the interval \( [a, b] \). It’s similar to adding up an infinite number of infinitesimally small line segments that represent our curve. With integration, we might cope with continuous motion just as easily as if we were counting apples in a basket. This step-by-step build-up from differential elements to the total arc length is the essence of the integral calculus part of our journey.