Area Between CurvesExpected Educational ResultsBloom’s TaxonomyArea Between CurvesInvestigation 02CC 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.

Use the embedded DESMOS graph below:

Explain why

.$\text{Area}\ge 0$ Identify the shaded region representing the area between

and the$x=y-2$ -axis on$y$ .$-1\le y\le 2]$ Identify the shaded region representing the area under

on$x={y}^{2}-4$ .$-1\le y\le 2]$ Which of the two shaded regions above is the larger shaded region? Explain.

Identify the shaded region representing the area \textbf{between}

and$x=y-2$ on$x={y}^{2}-4$ .$-1\le y\le 2]$ Solve the equation for

:$y$ $y-2={y}^{2}-4$ - Explain why there
**must**be two solutions to the above equation. - Compare the solutions to the above equation to the
-coordinates of the intersections of$y$ and$x=y-2$ . What do you notice? Explain.$x={y}^{2}-4$

- Explain why there
Identify

.${\int}_{-1}^{2}(y-2){\textstyle \phantom{\rule{0.167em}{0ex}}}dy$ Identify

.${\int}_{-1}^{2}({y}^{2}-4){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ Which of the above two integrals is the larger value? Explain.

Compare

with${\int}_{-1}^{2}(y-2){\textstyle \phantom{\rule{0.167em}{0ex}}}dx-{\int}_{-1}^{2}({y}^{2}-4){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ .${\int}_{-1}^{2}(y-2-({y}^{2}-4)){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ What do you notice? Explain.

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