91Ó°ÊÓ

Evaluate the sums. $$ \text { a. }sum_{k=1}^{10} k \quad \text { b. } \sum_{k=1}^{10} k^{2} \quad \text { c. } \sum_{k=1}^{10} k^{3} $$

Evaluate the integrals $$ \int_{1}^{-1}(r+1)^{2} d r $$

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. $$\begin{array}{|c|c|c|c|c|c|}\hline \text { Time (h) } & {0} & {1} & {2} & {3} & {4} \\ \hline \text { Leakage (gal/h) } & {50} & {70} & {97} & {136} & {190} \\ \hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline \text { Time (h) } & {5} & {6} & {7} & {8} \\\ \hline \text { Leakage (gallh) } & {265} & {369} & {516} & {720} \\\ \hline\end{array}$$ 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 gal \(/ \mathrm{h}\) after the first 8 hours. If the tanker originally contained \(25,000\) gal of oil, approxi- mately how many more hours will elapse in the worst case before all the oil has spilled? In the best case?

In Exercises \(15-22,\) graph the integrands and use known area formulas to evaluate the integrals. $$ \int_{-2}^{1}|x| d x $$

Evaluate the integrals in Exercises \(17-50\) $$ \int \theta \sqrt[4]{1-\theta^{2}} d \theta $$

Use the Substitution Formula in Theorem 7 to evaluate the integrals. $$ \int_{0}^{\pi / 4}(1-\sin 2 t)^{3 / 2} \cos 2 t d t $$

Evaluate the integrals in Exercises \(17-50\) $$ \int 3 y \sqrt{7-3 y^{2}} d y $$

In Exercises \(15-22,\) graph the integrands and use known area formulas to evaluate the integrals. $$ \int_{-1}^{1}(1-|x|) d x $$

Evaluate the sums. $$ \text { a. }sum_{k=1}^{13} k \quad \text { b. } \sum_{k=1}^{13} k^{2} \quad \text { c. } \sum_{k=1}^{13} k^{3} $$

Evaluate the integrals $$ \int_{-\sqrt{3}}^{\sqrt{3}}(t+1)\left(t^{2}+4\right) d t $$