91Ó°ÊÓ

Parametric equations describe a set of related quantities as expressions dependent on one or more independent parameters, usually denoted by letters like \( t \) or \( s \). In the context of the arc length problem, we have parametric equations for three dimensions: \( x = \cos^3(t) \), \( y = \sin^3(t) \), and a constant \( z = 2 \). Each function describes a component of the position in space.

• These equations allow us to represent curves that may be complex or even impossible to describe with a single Cartesian equation.
• The parameter \( t \) varies over an interval, providing a complete description of the curve's shape and position as \( t \) changes.
• In three dimensional space, we describe the position with \( (x(t), y(t), z(t)) \), tracing the curve as \( t \) varies.

By using parametric equations, we gain flexibility and power in analyzing and manipulating curves in higher dimensions, which is essential for calculating properties like arc lengths.

Differentiation is a fundamental tool in calculus, used to compute the rate of change of one quantity with respect to another. In the context of computing the arc length of a parametric curve, differentiation helps express how each coordinate changes as the parameter \( t \) changes.

• For the given parametric equations, the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) describe how the \( x \) and \( y \) coordinates change as \( t \) varies.
• Since \( z = 2 \) is constant, \( \frac{dz}{dt} \) equals zero, indicating no change in the \( z \)-coordinate.

Differentiation is critical because the arc length formula involves these derivatives to quantify the total change in path: \[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} \, dt \] This expression measures the infinitesimal distance traveled along the curve, summing these tiny distances from \( t = 0 \) to \( t = \pi/2 \), which provides the total arc length.

An integral accumulates quantities across an interval, acting as a continuous sum. In calculating the arc length, we use integration to total the infinitesimal distances calculated from our differentiated parametric equations.

Firstly, integrate the expression derived from the derivatives: \[ L = \int_{0}^{\pi/2} 3 \cos(t)\sin(t) \, dt \] The integral effectively adds up all the small linear segments over the specified interval of \( t \).

• In our problem, the integral reveals the nature of the curve by evaluating the cumulative distance along the curve without looking at it as a straight line.
• Substitution techniques simplify integration, here transforming \( 3\cos(t)\sin(t) \) into a trigonometric identity: \( \sin(2t) = 2\sin(t)\cos(t) \).

After performing this integration and applying the limits \( t = 0 \) to \( t = \pi/2 \), we found that the arc length is precisely 3 units. This shows the powerful synergy of parametric equations, differentiation, and integration in analyzing curves in space.