91Ó°ÊÓ

In mathematics, parametric equations offer a dynamic way to describe lines or curves through a set of equations that express the coordinates of the curve as functions of a parameter. In our case, the curve \( C \) is defined using parametric equations for \( x \), \( y \), and \( z \) based on the parameter \( t \), specifically:

• \( x = 4\cos t \)
• \( y = 4\sin t \)
• \( z = 3t \)

Here, \( t \) varies from \( 0 \) to \( 2\pi \), producing a complete path along the curve. Parametric methods are particularly useful because they allow us to handle curves that may not be easily described in a standard Cartesian form. By considering each component's function of \( t \), we can capture complex movements, like spiral or circular paths that appear frequently in physical applications.

Vector calculus extends the ideas of standard calculus to fields that describe vector quantities. These can include vector fields where each point in space is linked not to a single value, but to a vector.
Our task involves using vector calculus to compute a line integral over a curve, \( C \), parameterized by \( \vec{r}(t) = \langle 4\cos t, 4\sin t, 3t \rangle \). A line integral like this is integral to understanding physical concepts such as work done by a force field.
To progress, compute the derivative \( \vec{r}'(t) \) which gives a vector that is tangent to the curve at each point:

• \( \vec{r}'(t) = \langle -4\sin t, 4\cos t, 3 \rangle \)

The magnitude of this derivative \( \left| \vec{r}'(t) \right| \) is crucial as it incorporates into the integral to scale our line integral by the length of the differential path segment, \( ds \). Calculating \( \left| \vec{r}'(t) \right| \) gives \( 5 \) simply through the Pythagorean Theorem since the trigonometric identity simplifies the process \( \sin^2 t + \cos^2 t = 1 \).

Helical curves are a type of three-dimensional spiral. In this instance, our parametric equations define a helix via \( x = 4\cos t \), \( y = 4\sin t \), and \( z = 3t \). These represent the characteristics of a helix, where:

• The \( x \) and \( y \) equations describe a circle of radius 4 in the plane \( z = 0 \).
• The \( z \) component, changing linearly with \( t \), adds vertical motion to the circular base.

Helices frequently appear in nature and technology, such as DNA structures or spring mechanics. Their appearance in mathematical models allows us to describe phenomena that involve rotation and translation in unison. The helical path leads our line integral, allowing us to track accumulations such as distance or field interactions along this spiraling journey.

The evaluation of line integrals is a key skill in mathematical and engineering contexts. Starting from \( \int_{C}(x^{2} + y^{2} + z^{2})\ ds \), you compute the value by substituting the parametric functions. After substituting, simplify to find \( x^{2} + y^{2} + z^{2} = 16 + 9t^{2} \).
Then, multiply by the magnitude of the derivative, \( \left| \vec{r}'(t) \right| = 5 \), transforming it into a standard integral:

• \( 5 \int_{0}^{2\pi} (16 + 9t^{2}) \ dt \)

Next, split and solve the integral:

• The integral of a constant: \( \int_{0}^{2\pi} 16 \ dt = 32\pi \)
• The quadratic integral: \( \int_{0}^{2\pi} 9t^2 \ dt = \frac{8\pi^3}{3} \)

Combining these results gives the final evaluated integral value: \( 160\pi + 40\pi^3 \). This process is not only mathematical but practical, representing cumulative effects along a pathway in physics and engineering.