91Ó°ÊÓ

Suppose that \(\sum_{k=1}^{n} a_{k}=-5\) and \(\sum_{k=1}^{n} b_{k}=6 .\) Find the values of a. \(\sum_{k=1}^{n} 3 a_{k}\) b. \(\sum_{k=1}^{n} \frac{b_{k}}{6} \quad\) c. \(\sum_{k=1}^{n}\left(a_{k}+b_{k}\right)\) d. \(\sum_{k=1}^{n}\left(a_{k}-b_{k}\right) \quad\) e. \(\sum_{k=1}^{n}\left(b_{k}-2 a_{k}\right)\)

Evaluate the integrals. $$\int \sqrt{3-2 s} d s$$

Use the Substitution Formula in Theorem 7 to evaluate the integrals. $$\int_{0}^{\pi / 6} \cos ^{-3} 2 \theta \sin 2 \theta d \theta$$

Evaluate the integrals. $$\int_{0}^{\pi / 8} \sin 2 x d x.$$

Suppose that \(\sum_{k=1}^{n} a_{k}=0\) and \(\sum_{k=1}^{n} b_{k}=1 .\) Find the values of a. \(\sum_{k=1}^{n} 8 a_{k} \quad\) b. \(\sum_{k=1}^{n} 250 b_{k}\) c. \(\sum_{k=1}^{n}\left(a_{k}+1\right)\) d. \(\sum_{k=1}^{n}\left(b_{k}-1\right)\)

Use the Substitution Formula in Theorem 7 to evaluate the integrals. $$\int_{\pi}^{3 \pi / 2} \cot ^{5}\left(\frac{\theta}{6}\right) \sec ^{2}\left(\frac{\theta}{6}\right) d \theta$$

Graph the integrands and use known area formulas to evaluate the integrals. $$\int_{-4}^{0} \sqrt{16-x^{2}} d x$$

Evaluate the integrals.$$\int_{-\pi / 3}^{-\pi / 4}\left(4 \sec ^{2} t+\frac{\pi}{t^{2}}\right) d t.$$

Evaluate the integrals. $$\int \frac{1}{\sqrt{5 s+4}} d s$$

Water pollution Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening as evidenced by the increased leakage each hour, recorded in the following table. $$\begin{array}{l|c|c|c|c|c|} \text { Time (h) } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Leakage (L/h) } & 50 & 70 & 97 & 136 & 190 \end{array}$$ $$\begin{array}{l|c|c|c|c|} \text { Time (h) } & 5 & 6 & 7 & 8 \\ \hline \text { Leakage (L/h) } & 265 & 369 & 516 & 720 \end{array}$$ a. Give an upper and a lower estimate of the total quantity of oil that has escaped after 5 hours. b. Repeat part (a) for the quantity of oil that has escaped after 8 hours. c. The tanker continues to leak \(720 \mathrm{L} / \mathrm{h}\) after the first 8 hours. If the tanker originally contained 25,000 L of oil, approximately how many more hours will elapse in the worst case before all the oil has spilled? In the best case?