Area Between CurvesExpected Educational ResultsBloom’s TaxonomyArea Between CurvesDefinitionsProcedureInvestigation 05Investigation 06CC 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.

For each of the following, use the embedded DESMOS graph to graph both functions and to find the intersections of the curves.

- Find the area between
and$y={x}^{2}-1$ .$y=2x+7$ - Find the area between
and$y=\sqrt{x}$ .$y=x$

- Find the area between
and$x={y}^{2}$ .$x=2y$ - Find the area between
and$x=-{y}^{2}-8y-15$ .$2x={y}^{2}+8y+6$

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