# 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 01

Use the embedded DESMOS graph below:

1. Identify the shaded region representing the area under $f\left(x\right)=\mathrm{sin}\left(\pi x\right)$$f(x)=\sin{(\pi x)}$ on $\left[0,0.5\right]$$[0, 0.5]$.
2. Identify ${\int }_{0}^{0.5}\mathrm{sin}\left(\pi x\right)\phantom{\rule{0.167em}{0ex}}dx$$\int_0^{0.5}{\sin{(\pi x)}\,dx}$.
3. Identify the shaded region representing the area under $g\left(x\right)=2x$$g(x)=2x$ on $\left[0,0.5\right]$$[0, 0.5]$.
4. Identify ${\int }_{0}^{0.5}2x\phantom{\rule{0.167em}{0ex}}dx$$\int_0^{0.5}{2x\,dx}$.
5. Identify the shaded region representing the area \textbf{between} $f\left(x\right)=\mathrm{sin}\left(\pi x\right)$$f(x)=\sin{(\pi x)}$ and $g\left(x\right)=2x$$g(x)=2x$ on $\left[0,0.5\right]$$[0, 0.5]$.
6. Compare ${\int }_{0}^{0.5}\mathrm{sin}\left(\pi x\right)\phantom{\rule{0.167em}{0ex}}dx-{\int }_{0}^{0.5}2x\phantom{\rule{0.167em}{0ex}}dx$$\int_0^{0.5}{\sin{(\pi x)}\,dx} - \int_0^{0.5}{2x\,dx}$ with ${\int }_{0}^{0.5}\left(\mathrm{sin}\left(\pi x\right)-2x\right)\phantom{\rule{0.167em}{0ex}}dx$$\int_0^{0.5}{\left(\sin{(\pi x)}-2x\right)\,dx}$.
7. What do you notice? Explain.