# 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 02

Use the embedded DESMOS graph below:

1. Explain why $\text{Area } \geq 0$.

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

3. Identify the shaded region representing the area under $x={y}^{2}-4$$x=y^2-4$ on $-1\le y\le 2\right]$$-1\leq y \leq 2]$.

4. Which of the two shaded regions above is the larger shaded region? Explain.

5. Identify the shaded region representing the area \textbf{between} $x=y-2$$x=y-2$ and $x={y}^{2}-4$$x=y^2-4$ on $-1\le y\le 2\right]$$-1\leq y \leq 2]$.

6. Solve the equation for $y$$y$: $y-2={y}^{2}-4$$y-2=y^2-4$

1. Explain why there must be two solutions to the above equation.
2. Compare the solutions to the above equation to the $y$$y$-coordinates of the intersections of $x=y-2$$x=y-2$ and $x={y}^{2}-4$$x=y^2-4$. What do you notice? Explain.
7. Identify ${\int }_{-1}^{2}\left(y-2\right)\phantom{\rule{0.167em}{0ex}}dy$$\int_{-1}^{2}{\left(y-2\right)\,dy}$.

8. Identify ${\int }_{-1}^{2}\left({y}^{2}-4\right)\phantom{\rule{0.167em}{0ex}}dx$$\int_{-1}^{2}{\left(y^2-4\right)\,dx}$.

9. Which of the above two integrals is the larger value? Explain.

10. Compare ${\int }_{-1}^{2}\left(y-2\right)\phantom{\rule{0.167em}{0ex}}dx-{\int }_{-1}^{2}\left({y}^{2}-4\right)\phantom{\rule{0.167em}{0ex}}dx$$\int_{-1}^{2}{\left(y-2\right)\,dx} - \int_{-1}^{2}{\left(y^2-4\right)\,dx}$ with ${\int }_{-1}^{2}\left(y-2-\left({y}^{2}-4\right)\right)\phantom{\rule{0.167em}{0ex}}dx$$\int_{-1}^{2}{\left(y-2-\left(y^2-4\right)\right)\,dx}$.

11. What do you notice? Explain.