MATH $181$ TECHNOLOGICAL COMPETENCY & CRITICAL REASONING

Problem Directions:

Consider $f(x)=xsinx$

A$.$ Estimate ${\displaystyle \underset{0}{\overset{3}{}}}xsinxdx$ with $L_{10}.$

B$.$ Estimate ${\displaystyle \underset{0}{\overset{3}{}}}xsinxdx$ with $R_{10}.$

C$.$ Estimate ${\displaystyle \underset{0}{\overset{3}{}}}xsinxdx$ with $M_{10}.$

D$.$ Which of the three estimates is the least accurate? Why?

E$.$ Find a subinterval $a,b$ of the interval $0,3$ for which $L_{10}$ would give an underestimate of

$,{\displaystyle \underset{a}{\overset{b}{}}}xsinxdx.$ Explain why you chose that interval and why $L_{10}isan$ underestimate $of$

${\displaystyle \underset{a}{\overset{b}{}}}xsinxdx$

F$.$ Provide an example of a "real$-$life" situation in which you would prefer that the estimate of an integral be an underestimate rather than an overestimate? Explain why the underestimate is desired.