A pharmaceutical researcher models the concentration of a therapeutic drug, CCC, in milligrams per litre (mg/L), present in a patient's bloodstream ttt hours after the initial dose using the function:
C(t)=26−2t+1,t≥0 C(t) = 2^{6 - \sqrt{2t+1}}, \quad t \ge 0 C(t)=26−2t+1,t≥0The table below shows corresponding values of ttt and C(t)C(t)C(t). The values of C(t)C(t)C(t) are given to 3 decimal places.
| ttt | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|
| C(t)C(t)C(t) | 8.000 | 5.258 | 3.673 | 2.671 | 2.000 |
Using the trapezium rule with all the values in the table:
obtain an estimate for the total drug exposure over the interval 4≤t≤124 \le t \le 124≤t≤12, given by ∫412C(t) dt\int_{4}^{12} C(t) \, dt∫412C(t)dt, giving your answer to 2 decimal places.
Using your answer to part (a) and making your method clear, estimate
(i) ∫41227−2t+1 dt\int_{4}^{12} 2^{7-\sqrt{2t+1}} \, dt∫41227−2t+1dt
(ii) ∫412(C(t)+5) dt\int_{4}^{12} (C(t) + 5) \, dt∫412(C(t)+5)dt
