Study Notes on Area Between Two Curves

AREA BETWEEN CURVES

• Definition: If $f(x) \ngeq g(x)$ for all $x$ in the interval $[a, b]$, then the area of the region between the graphs of $f$ and $g$ is given by:   - $A = \int_{a}^{b} [f(x) - g(x)] \, dx$
    This formula represents the integral of the difference between the two functions across the specified interval, yielding the area between their curves.

STEPS TO FIND THE AREA BETWEEN CURVES

1. Graphing:
      - Sketch the graphs of $f(x)$ and $g(x)$ to visually determine points of intersection. Identify which function is larger or above the other throughout the interval.

2. Identifying Larger Function:
      - If $f(x)$ is consistently larger than $g(x)$ in the interval, proceed with:
        - $A = \int_{a}^{b} [f(x) - g(x)] \, dx$

3. Handling Intersections:
      - If neither function is always above, split the integral into pieces based on the points of intersection and sum the areas:
     - $A = \int_{a}^{c} [f(x) - g(x)] \, dx + \int_{c}^{b} [g(x) - f(x)] \, dx$, where $c$ is the x-coordinate of their intersection.

General Implications:

• The method of finding the area between curves emphasizes the importance of integration techniques and function analysis, applicable in theoretical and practical scenarios, including computing areas in various fields.