Area Between Curves

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

• Section 6.1 – Area Between Curves

Expected Educational Results

• 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.

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.

Area Between Curves

Investigation 03

Use the embedded DESMOS graph below:

1. Explain why the area between $y=x^3-x+2$$y=x^3-x+2$ and $y=3x+2$$y=3x+2$ must $\text{Area }>0$$\text{Area } > 0$.

2. Solve the equation for $x$$x$: $x^3-x+2=3x+2$$x^3-x+2=3x+2$.

   1. Explain why there must be three (3) solutions to the above equation.
   2. Compare the solutions to the above equation to the $x$$x$-coordinates of the intersections of $y=x^3-x+2$$y=x^3-x+2$ and $y=3x+2$$y=3x+2$. What do you notice? Explain.

3. Explain why ${\int }_{-2}^2(x^3-x+2-(3x+2))dx$$\int_{-2}^{2}{\left(x^3-x+2-\left(3x+2\right)\right)\,dx}$ does not represent the area between $y=x^3-x+2$$y=x^3-x+2$ and $y=3x+2$$y=3x+2$.

4. Compare ${\int }_{-2}^0(x^3-x+2-(3x+2))dx+{\int }_0^2(3x+2-(x^3-x+2))dx$$\int_{-2}^{0}{\left(x^3-x+2-\left(3x+2\right)\right)\,dx} + \int_{0}^{2}{\left(3x+2-\left(x^3-x+2\right)\right)\,dx}$ with ${\int }_{-2}^2|x^3-x+2-(3x+2)|dx$$\int_{-2}^{2}{\left|x^3-x+2-\left(3x+2\right)\right|\,dx}$.

5. What do you notice? Explain.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

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