AP Calculus AB TOPIC 8.4 Finding the Area Between Curves Expressed as Functions of x

Please subscribe!    / nickperich   *Learning Objective: CHA-5.A* *Calculate areas in the plane using the definite integral.* This objective focuses on finding the area of regions enclosed by curves and boundaries on a graph. Using definite integrals, students learn to evaluate the area between functions by integrating the difference between their values over a given interval. --- *Essential Knowledge: CHA-5.A.1* *Areas of regions in the plane can be calculated with definite integrals.* Key Concepts: 1. **Area Between Two Curves**: For functions \( f(x) \) and \( g(x) \) over an interval \( [a, b] \): If \( f(x) \) is above \( g(x) \): \[ \text{Area} = \int_a^b \left[ f(x) - g(x) \right] dx \] Adjust for regions where curves intersect or switch dominance. 2. **Geometric Approach**: The definite integral computes the net signed area under a curve. For regions requiring total positive area, functions must be evaluated appropriately. 3. **Vertical or Horizontal Integration**: Typically, integration is performed with respect to \( x \), but when the region is better expressed in terms of \( y \), integration with respect to \( y \) can be used: \[ \text{Area} = \int_{y=c}^{y=d} \left[ f(y) - g(y) \right] dy \] 4. **Boundaries Defined by Functions and Lines**: Regions may be bounded by curves, vertical/horizontal lines, or intersections of functions. Understanding these boundaries is crucial for setting up integrals. --- *Examples* 1. *Area between \( y = x^2 \) and \( y = x \) over \( x \in [0, 1] \):* \[ \text{Area} = \int_0^1 \left( x - x^2 \right) dx \] 2. *Area under \( y = \sin(x) \) from \( x = 0 \) to \( x = \pi \):* \[ \text{Area} = \int_0^\pi \sin(x) dx \] These examples illustrate how definite integrals are used to calculate areas enclosed by curves and boundaries, reinforcing the importance of correctly identifying upper and lower functions over the interval. I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study