The concentration CCC (in mg/L) of a metabolic byproduct in a bioreactor is monitored over a 2-hour period. The rate of change of concentration is modeled by the function C(t)=81+2tC(t) = \frac{8}{\sqrt{1 + 2^t}}C(t)=1+2t8 for 0≤t≤20 \le t \le 20≤t≤2, where ttt is measured in hours.
A researcher uses the trapezium rule with 5 ordinates (4 strips) to find an approximation for the total exposure (the area under the curve) over this interval. The values required for this approximation are shown in the table below.
| ttt | 0 | 0.5 | 1 | 1.5 | 2 |
|---|---|---|---|---|---|
| CCC | 5.65685 | 5.14875 | 4.61880 | 4.08865 | 3.57771 |
Use the trapezium rule with all 5 ordinates from the table to find an approximate value for the total exposure over the interval 0≤t≤20 \le t \le 20≤t≤2. Give your answer to four decimal places.
Using your answer to part (a), deduce an estimate for ∫02241+2t dt\int_{0}^{2} \frac{24}{\sqrt{1 + 2^t}} \, dt∫021+2t24dt.