A temperature sensor records the temperature T (in °C) of a liquid every minute during a 10-minute cooling process. The data are modeled by a differentiable function T that satisfies T(t)=20+60e−0.2t for 0≤t≤10, where t is measured in minutes.
To estimate the total heat lost (in degree-minutes) during the cooling period, an engineer considers the Riemann sum obtained by partitioning the interval into n subintervals of equal length and evaluating T at the right-hand endpoints.
(a) Write the limit of Riemann sums that represents the exact total heat lost from t=0 to t=10.
(b) Express the limit in part (a) as a definite integral and evaluate it exactly.
(c) Compute the average temperature of the liquid over the ten-minute interval.
(d) Let Q(x)=∫0x(T(t)−20)dt for 0≤x≤10. Interpret Q(10) in the context of the problem and determine Q′(x), stating appropriate units for both quantities.