Area was used as a motivation for developing the definition of Riemannian Integration. Now we develop techniques for calculating more advanced areas.
Subsection4.7.1Restricting Integrals for Areas
Integrals can be used to calculate areas. However integrals are not area by definition.
Activity4.7.1.Integrals which Represent Areas.
The goal of this activity is to determine when an integral represents an area.
(a)
Evaluate \(\int_0^{2\pi} \sin(\theta) d\theta \)
(b)
Evaluate \(\int_{-1}^{1} x^3 dx \)
(c)
Evaluate \(\int_{-1}^{1} 1-x^2 dx \)
(d)
Evaluate \(\int_0^{1} e-e^x dx \)
(e)
Which of the integrals above are reasonably interpreted as area and why?
Subsection4.7.2Setting up Area Calculations
Activity4.7.2.How to Setup Area Integrals.
The goal of this activity is to learn how to setup integrals for areas enclosed by curves. Enclosed can be thought of as able to hold water if you poured it into the curves. For example a circle encloses an area. A line cannot enclose an area.
(a)
Setup an integral to calculate the area enclosed between the x-axis and \(f(x)=1-(x-1)^2 \text{.}\)
(b)
Setup an integral to calculate the area enclosed between the x-axis and \(g(x)=(x-1)^2-1 \text{.}\)
(c)
Setup an integral to calculate the area enclosed between \(f(x)\) and \(g(x)\text{.}\)
(d)
Sketch \(f(x)=x^3-x^2-x+1\) and \(g(x)=1-x^2\) on the same axes.
(e)
Identify the area enclosed between these two curves.
(f)
Setup the calculation for the area enclosed between \(f(x)\) and \(g(x)\text{.}\)
