91Ó°ÊÓ

Parametric equations provide a way to describe a curve by expressing the coordinates of the points on the curve as functions of a third variable, usually time denoted as \( t \). In our exercise, we have \( x = \cos t \) and \( y = 2 + \sin t \). Here, \( t \) varies between \( 0 \) and \( 2\pi \), tracing a path on the xy-plane. Parametric equations are often used when dealing with curves that are difficult to express with a single function of \( x \) or \( y \) alone.

• Independent parameter \( t \): Enables the tracking of a curve within a specified range.
• Coordinate functions: Define the \( x \) and \( y \) positions as \( t \) changes.

These equations are particularly useful in calculus for dealing with more complex motion and geometry, allowing us to explore transformations and rotations spherically by altering \( x \) and \( y \) independently.

Integral calculus serves as a comprehensive tool for determining accumulations, such as areas under curves or surface areas. The integral is a fundamental concept, which in this exercise is used to calculate the surface area created by revolving a curve around an axis. Let's break this down:

• The Definite Integral: Provides the total accumulation of quantities over a finite interval \([a, b]\).
• Integral Expression: In our context, \( \int_0^{2\pi} 2\pi (2 + \sin t) \, dt \) signifies the sum of infinitesimally small sections of the surface being revolved, gathered over the interval from \( t = 0 \) to \( t = 2\pi \).

The definite integral effectively computes this total by analyzing the curve's behavior over this range, integrating elements like length and curvature, and providing results that are otherwise complex to derive.

Calculating the surface area of a revolution requires a careful application of the formula for parametric curves. When the curve \( x = \cos t \) and \( y = 2 + \sin t \) is revolved around the x-axis, we employ the formula:\[ S = \int_a^b 2\pi g(t) \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } \, dt \]This formula articulates how each small segment of the curve forms part of the revolution's surface.

• Contribution of each segment: The part \( 2\pi g(t) \) accounts for the circumference of the circle formed by the segment.
• Expression for arc length: The term \( \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } \) reflects the infinitesimal arc length involved.

Understanding these pieces and their collective role is crucial. Substituting the derivatives \( \frac{dx}{dt} = -\sin t \) and \( \frac{dy}{dt} = \cos t \) simplifies the complexity, making it possible to solve the integral and thus determine the surface area \( S = 8\pi^2 \). This approach exemplifies the powerful intersection between calculus and geometric visualization.