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(x)=\sin (\pi x)$$f(x)=\sin{(\pi x)}$ on $[0,0.5]$$[0, 0.5]$.
2. Identify ${\int }_0^{0.5}\sin (\pi x)dx$$\int_0^{0.5}{\sin{(\pi x)}\,dx}$.
3. Identify the shaded region representing the area under $g(x)=2x$$g(x)=2x$ on $[0,0.5]$$[0, 0.5]$.
4. Identify ${\int }_0^{0.5}2xdx$$\int_0^{0.5}{2x\,dx}$.
5. Identify the shaded region representing the area \textbf{between} $f(x)=\sin (\pi x)$$f(x)=\sin{(\pi x)}$ and $g(x)=2x$$g(x)=2x$ on $[0,0.5]$$[0, 0.5]$.
6. Compare ${\int }_0^{0.5}\sin (\pi x)dx-{\int }_0^{0.5}2xdx$$\int_0^{0.5}{\sin{(\pi x)}\,dx} - \int_0^{0.5}{2x\,dx}$ with ${\int }_0^{0.5}(\sin (\pi x)-2x)dx$$\int_0^{0.5}{\left(\sin{(\pi x)}-2x\right)\,dx}$.
7. What do you notice? Explain.

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