91Ó°ÊÓ

Parametric equations allow you to express a curve using a parameter, typically represented as "t". These equations define the x and y coordinates uniquely in terms of "t". In this exercise, the parametric equations are given by

• \( x = 1 - t \)
• \( y = 2 + 3t \)

These two equations describe a straight line in a plane. Rather than expressing \( y \) directly in terms of \( x \), parametric equations provide a versatile way to portray complex curves beyond simple functions.
In this specific problem, the parameter "t" ranges from \(-\frac{2}{3} \) to \(1\), which identifies which portion of the line you're interested in studying.
Understanding these equations can unlock insights into the geometry and behavior of curves and allow for more flexible problem-solving techniques than traditional Cartesian functions.

Integral calculus plays a vital role when it comes to measuring properties such as area, volume, and in this case, the length of curves. The length of a curve defined via parametric equations can be determined using the integral curve length formula:\[L = \int_a^b \sqrt{ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]
This formula essentially adds up infinitely small line segments dictated by the changes in \( x \) and \( y \) as dictated by their derivatives, forming the entire curve over the interval \([a, b]\).
In the given solution, the integral of \( \int_{-2/3}^{1} \sqrt{10} \ dt \) calculates the total length by considering all the small changes over "t" from \(-\frac{2}{3} \) to \(1\). Calculus integrates these components to provide a collective measure, demonstrating the power of integration in solving real-world problems such as finding the arc length.

Derivatives represent the rate at which things change. For parametric equations, we compute the derivative of \( x \) and \( y \) with respect to the parameter "t", which provides details on how each coordinate changes as "t" changes.
In the provided exercise, the derivatives are:

• \( \frac{dx}{dt} = -1 \)
• \( \frac{dy}{dt} = 3 \)

These derivatives tell us the direction and speed of change for the curve in the x and y directions. For the length of the curve, they are squared, summed, and square-rooted in the curve length formula. Derivatives transform raw positions into information about how a point moves along the curve, helping us piece together the whole shape and path of the curve.