# 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 $\text{Area } > 0$.

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$$y=3x+2$. What do you notice? Explain.
3. Explain why ${\int }_{-2}^{2}\left({x}^{3}-x+2-\left(3x+2\right)\right)\phantom{\rule{0.167em}{0ex}}dx$$\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}\left({x}^{3}-x+2-\left(3x+2\right)\right)\phantom{\rule{0.167em}{0ex}}dx+{\int }_{0}^{2}\left(3x+2-\left({x}^{3}-x+2\right)\right)\phantom{\rule{0.167em}{0ex}}dx$$\int_{-2}^{0}{\left(x^3-x+2-\left(3x+2\right)\right)\,dx} + \int_{0}^{2}{\left(3x+2-\left(x^3-x+2\right)\right)\,dx}$ with ${\int }_{-2}^{2}|{x}^{3}-x+2-\left(3x+2\right)|\phantom{\rule{0.167em}{0ex}}dx$$\int_{-2}^{2}{\left|x^3-x+2-\left(3x+2\right)\right|\,dx}$.

5. What do you notice? Explain.