Calculus: Section 6.1 - Area Between Curves

Objective Statement: The primary goal of this lesson is to enable students to find the area between curves using definite integrals.
Temporal Reference: The lesson is dated as 11/19.
Topic Identification: This material covers Section 6.1 of the curriculum, titled "Area Between Curves."

Finding the area of a region bounded by two functions requires the use of the definite integral. The area ( $A$ ) is fundamentally the limit of the sum of the areas of approximating rectangles.
Integration with Respect to $x$ : When the boundaries are defined as functions of $x$ , the area between two curves $f(x)$ and $g(x)$ from $a$ to $b$ is calculated using: - $A = \int_a^b (f(x) - g(x))\,dx$ - In this context, $f(x)$ represents the upper function (the "top" curve) and $g(x)$ represents the lower function (the "bottom" curve).
Integration with Respect to $y$ : When the boundaries are defined as functions of $y$ (horizontal orientation), the area ( $A$ ) is calculated between the right-hand curve ( $f(y)$ ) and the left-hand curve ( $g(y)$ ) over the interval $[c, d]$ : - $A = \int_c^d (f(y) - g(y))\,dy$ - This approach is often necessary if the functions are easier to express as $x = f(y)$ or if the region is bounded naturally by horizontal lines.

Based on the documented transcript entries from page 1 and page 2, the following numerical values and functional components are identified:

Numerical Constants and Thresholds: The transcript contains high-frequency appearances of specific numbers, including: - $161$ , $4$ , $12$ , $3$ , $1$ , $7$ , $48$ , $6$ , $0$ , $1$ , $11$ , $2$ , and $21$ . - Large numerical sequences mentioned include $513$ and $1452$ .
Variables and Symbols: Functional relationships involve variables $x$ , $y$ , and several instances of square notation indicated by "2" (likely representing exponents like $y^2$ ).
Integral Process Indicators: The term "dy" on Page 2 explicitly confirms that the lesson includes instances of integrating with respect to the vertical axis ( $y$ -axis).
Specific Expressions from Transcript (Page 2): - A sequence of functional fragments is noted: 2y 4 y y F4 y 2 y F 5 4 y 2y dy. - This suggests the calculation of an area between a linear function of $y$ and a quadratic function of $y$ .

Identify the Boundaries: Determine the points of intersection (where $f(x) = g(x)$ ) to establish the limits of integration ( $a$ and $b$ ).
Determine the Orientation: Decide whether to integrate with respect to $x$ ( $dx$ ) or $y$ ( $dy$ ). - If curves are given as $y = f(x)$ , use $dx$ . - If curves are given as $x = f(y)$ , use $dy$ .
Identify Upper/Lower or Right/Left: - For $dx$ , determine which function has higher $y$ -values over the interval. - For $dy$ , determine which function has higher $x$ -values (farthest to the right) over the interval.
Set Up the Definite Integral: Substitute the functions into the subtraction formula ( $f - g$ ).
Evaluate the Integral: Use the Fundamental Theorem of Calculus to find the total area.

Reference Location: The transcript explicitly points to Pg 488 for further study and context regarding these specific problems and definitions.
Problem Set Correlation: The numbers $21$ , $4$ , and $2$ mentioned on Page 2 appear to be related to specific example problems or exercises located on the aforementioned book page.