The rate of accumulation of a specific pollutant in a coastal reservoir, RRR, measured in kg/hour, is modeled by the function R(t)=3t3t2+9R(t) = \frac{3t^3}{t^2+9}R(t)=t2+93t3 for 0≤t≤80 \le t \le 80≤t≤8, where ttt represents the time in hours after a filtration system failure.
A researcher approximates the total mass of pollutant accumulated during this 8-hour period, MMM, using the trapezium rule with n=4n = 4n=4 equal intervals. (i) State the number of ordinates that the researcher uses. (ii) Calculate the approximation for MMM that the researcher should obtain. Give your answer correct to two decimal places.
Show that the exact mass of pollutant accumulated, given by M=∫08R(t) dtM = \int_{0}^{8} R(t) \, dtM=∫08R(t)dt, is exactly 96−272ln(739)96 - \frac{27}{2}\ln\left(\frac{73}{9}\right)96−227ln(973). Fully justify your answer.
Explain what would happen to the researcher's approximation in part (a)(ii) as n→∞n \to \inftyn→∞.