#Study Notes: Area Between Two Curves

#Table of Contents

1. Introduction
2. Finding the Area Between Two Curves in Terms of $x$
   • Steps for Calculation
   • Example Problem
3. Finding the Area Between Two Curves in Terms of $y$
   • Steps for Calculation
   • Example Problem
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways
7. Exam Strategy

#Introduction

In calculus, finding the area between two curves is a common problem that requires understanding how to set up and evaluate definite integrals. This guide will help you master this topic by breaking down the steps and providing illustrative examples.

#Finding the Area Between Two Curves in Terms of $x$

#Steps for Calculation

1. Identify the Curves and Bounds:

   • Consider the curves $y = f(x)$ and $y = g(x)$.
   • Identify the interval $[a, b]$ over which you want to find the area.

2. Set Up the Integral:

   • Ensure that $f(x) \ge g(x)$ over the interval $[a, b]$.
   • Calculate the area as: $$\text{Area} = \int_{a}^{b} (f(x) - g(x)) , dx$$

3. Evaluate the Integral:

   • Compute the definite integral to find the area between the curves.

#Example Problem

Find the area of the region enclosed by the curves $y = 2x^2 - 4x + 2$ and $y = -2x^2 + 8x - 6$.

1. Find the Points of Intersection:
   • Set the equations equal to each other and solve for $x$: $$2x^2 - 4x + 2 = -2x^2 + 8x - 6$$ Simplify and solve...