Area Between Curves

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Bloom’s Taxonomy for different levels of understanding
Figure 1.1: Bloom's Taxonomy

Area Between Curves

Definitions

Area Between Two Curves – 1:

The area A of the region bounded by the curves y=f(x) and y=g(x) where f and g are continuous and f(x)g(x) for all x in [a,b] is A=ab(f(x)g(x))dx

Area Between Two Curves – 2

The area A of the region bounded by the curves y=f(x) and y=g(x) where f and g are continuous for all x in [a,b] but f(x)g(x) or g(x)f(x) for all x on [a,b] is A=ab|f(x)g(x)|dx

Area Between Two Curves – 3

The area A of the region bounded by the curves x=f(y) and x=g(y) where f and g are continuous and f(y)g(y) for all cyd is A=cd(f(y)g(y))dy

Area Between Two Curves – 4

The area A of the region bounded by the curves x=f(y) and x=g(y) where f and g are continuous for all cyd but and f(y)g(y) or g(y)f(y) for all cyd is A=cd|f(y)g(y)|dy

Procedure

  1. Sketch the curves

  2. Identify the region enclosed by the curves

  3. Sketch one (1) typical rectangle within the region where every potential rectangle starts and ends at different curves. There are two potential rectangles to draw:

    1. Vertical rectangle [the integral will be in terms of the x-variable only]
    2. Horizontal rectangle [the integral will be in terms of the y-variable only]
  4. Find intersection(s) of curves [only the x- or y-coordinate of the intersections is needed, depending on the typical rectangle]

  5. Set up integral in one of the following ways:

    1. ab(f(x)g(x))dx, where f(x)g(x), x=a is the smaller x-coordinate of the intersection or smaller given value, and x=b is the larger x-coordinate of the intersection or larger given value;
    2. ab(f(y)g(y))dy, where f(y)g(y) [i.e., f(y) is to the right of g(y)], y=a is the smaller y-coordinate of the intersection or smaller given value, and y=b is the larger y-coordinate of the intersection or larger given value;
  6. If there are more than two intersections, then you will need to set up more than one integral and add the areas.

  7. Evaluate the integral.

Investigation 08

  1. Find the area enclosed by y=x2+3x, y=4x2, and y=xx24.

Homework

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Last Modified: Thursday, 17 September 2020 2:48 EDT