91Ó°ÊÓ

Parametric equations are a powerful way to describe shapes and curves in the plane. Instead of using just one equation involving \(x\) and \(y\), parametric equations express both \(x\) and \(y\) as functions of a third variable, commonly \(t\), called the parameter. This parameter often represents time or another changing quantity that determines the position of a point on a curve.
In the given exercise, the pair of equations \(x=3\cos t\) and \(y=3\sin t+1\) describe a circle with a radius of 3, centered at the point (0,1). As the parameter \(t\) varies from \(0\) to \(2\pi\), the point (x,y) traces a complete circle in the plane.
Using parametric equations can make it easier to work with curves involving circles, ellipses, and other complex shapes. They allow for seamless differentiation and integration, which are essential for finding properties like arc length, as shown in this problem.

Integration is a fundamental mathematical tool used to calculate many properties, including areas under curves and arc lengths. It allows us to sum up infinitesimally small quantities to find a total effect or quantity over an interval.
In our problem, once the derivatives are squared and summed, we need to find its integral to compute the arc length. The arc length \(L\) for a curve described by parametric equations \(x(t)\) and \(y(t)\) over the interval \(t=a\) to \(t=b\) is given by:

• \(L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\)

Here, the integral collects the infinitesimal lengths along the curve from start to finish. It's noted in our solution that integrating \(3\) over the interval \([0, 2\pi]\) yields \(6\pi\), giving us the total arc length of the circle.

Trigonometric identities are equations involving trigonometric functions that are true for all values where both sides of the equality are defined. They are extremely useful in simplifying calculations and solving problems involving trigonometric functions.
One of the most fundamental identities used in this problem is the Pythagorean Identity, which states:

• \(\sin^2 t + \cos^2 t = 1\)

This identity was used to simplify the expression \(9\sin^2 t + 9\cos^2 t\) to \(9\), aiding in the calculation of the arc length. Understanding and mastering these identities make manipulating and simplifying trigonometric expressions much easier, especially when involved in integration and differentiation tasks as seen here.