91Ó°ÊÓ

Which of the following express \(1+2+4+8+16+32\) in sigma notation? $$\begin{array}{lll} \text { a. } \sum_{k=1}^{6} 2^{k-1} & \text { b. } \sum_{k=0}^{5} 2^{k} & \text { c. } \sum_{k=-1}^{4} 2^{k+1} \end{array}$$

Which of the following express \(1-2+4-8+16-32\) in sigma notation? a. \(\sum_{k=1}^{6}(-2)^{k-1}\) b. \(\sum_{k=0}^{5}(-1)^{k} 2^{k}\) c. \(\sum_{k=-2}^{3}(-1)^{k+1} 2^{k+2}\)

Suppose that \(f\) is integrable and that \(\int_{0}^{3} f(z) d z=3\) and \(\int_{0}^{4} f(z) d z=7 .\) Find a. \(\int_{3}^{4} f(z) d z\) b. \(\int_{4}^{3} f(t) d t\)

Use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four sub intervals of equal length and evaluating \(f\) at the sub interval midpoints. $$f(x)=1 / x \quad \text { on } \quad[1,9]$$ (Graph cant copy)

Graph the integrands and use known area formulas to evaluate the integrals. $$\int_{-3}^{3} \sqrt{9-x^{2}} d x$$

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). $$\int_{0}^{\pi / 6} \cos ^{-3} 2 \theta \sin 2 \theta d \theta$$

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

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). $$\int_{-\pi / 2}^{\pi / 2} \frac{2 \cos \theta d \theta}{1+(\sin \theta)^{2}}$$

Find the norm of the partition \(P=\\{0,1.2,1.5,2.3,2.6,3\\}\).

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=x^{2}+1\) over the interval [0,3]