Area Between CurvesExpected Educational ResultsBloom’s TaxonomyArea Between CurvesDefinitionsProcedureInvestigation 08HomeworkCC BY-NC-SA 4.0

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

**Section 6.1**– Area Between Curves

**Objective 10–1**: I can set up the integral to compute the area of the region bounded by two or more curves.**Objective 10–2**: I can evaluate the integral to compute the area of the region bounded by two or more curves using FTC-II.**Objective 10–3**: I can use technology to estimate the area of the region bounded by two or more curves using FTC-II.

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.

**Area Between Two Curves – 1**:

The area

**Area Between Two Curves – 2**

The area

**Area Between Two Curves – 3**

The area

**Area Between Two Curves – 4**

The area

Sketch the curves

Identify the region enclosed by the curves

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:*Vertical*rectangle [the integral will be in terms of the -variable only]$x$ *Horizontal*rectangle [the integral will be in terms of the -variable only]$y$

Find intersection(s) of curves [only the

-$x$ **or** -coordinate of the intersections is needed, depending on the typical rectangle]$y$ Set up integral in one of the following ways:

, where${\int}_{a}^{b}(f(x)-g(x))dx$ ,$f(x)\ge g(x)$ is the smaller$x=a$ -coordinate of the intersection or smaller given value, and$x$ is the larger$x=b$ -coordinate of the intersection or larger given value;$x$ , where${\int}_{a}^{b}(f(y)-g(y))dy$ [i.e.,$f(y)\ge g(y)$ is to the right of$f(y)$ ],$g(y)$ is the smaller$y=a$ -coordinate of the intersection or smaller given value, and$y$ is the larger$y=b$ -coordinate of the intersection or larger given value;$y$

If there are more than two intersections, then you will need to set up more than one integral and add the areas.

Evaluate the integral.

- Find the area enclosed by
,$y={x}^{2}+3x$ , and$y=\frac{4}{{x}^{2}}$ .$y=x-\frac{{x}^{2}}{4}$

- At this time, you should be able to complete the following assignments:
**Section 6.1**: # 1, 3, 5, 7, 9, 11, 13, 15, 17, 23, 27, 37, 39, 41, 43, 47, 48, 49, 53.

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