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David Brown

David Brown

6 min read

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Study Guide Overview

This guide covers calculating the area between two curves using definite integrals. It explains how to find the area with respect to both x and y axes. It includes steps for identifying intersection points, setting up the integral, and evaluating it. The guide provides example problems, practice questions, a glossary, and exam strategies.

Study Notes: Area Between Two Curves

Table of Contents

  1. Introduction
  2. Finding the Area Between Two Curves in Terms of xx
  3. Finding the Area Between Two Curves in Terms of yy
  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 xx

Steps for Calculation

  1. Identify the Curves and Bounds:

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

    • Ensure that f(x)g(x)f(x) \ge g(x) over the interval [a,b][a, b].
    • Calculate the area as: Area=ab(f(x)g(x)),dx\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.
Key Concept

It's crucial to have the function that is "above" the other inside the integral first to ensure the integrand is always non-negative.

Example Problem

Find the area of the region enclosed by the curves y=2x24x+2y = 2x^2 - 4x + 2 and y=2x2+8x6y = -2x^2 + 8x - 6.

  1. Find the Points of Intersection:
    • Set the equations equal to each other and solve for xx: 2x24x+2=2x2+8x62x^2 - 4x + 2 = -2x^2 + 8x - 6 Simplify and solve...