Activity $4\mathrm{.}2\mathrm{.}3.$ Suppose that an object moving along a straight line path has its velocity in feet per second at time $t$ in seconds given by $v(t)=\frac{2}{9}(t-3{)}^2+2.$

a$.$ Carefully sketch the region whose exact area will tell you the value of the distance the object traveled on the time interval $2t5.$

b$.$ Estimate the distance traveled on $2,5$ by computing $L_4,R_4,$ and $M_4.$

c$.$ Does averaging $L_4$ and $R_4$ result in the shime value as $M_4?$ If not, what do you think the average of $L_4$ and $R_4$ measures?

d$.$ For this question, think about an arbitrary, function $f,$ rather than the particular function $v$ given above. If $f$ is positive and increasing on $a,b,$ will $L_n$ over$-$estimate or under$-$estimate the exact area under $f$ on $a,b?$ Will $R_n$ over$-$ or under$-$estimate the exact area under $f$ on $a,b?$ Explain.