Chapter 5

Evaluate the integrals. $$\int_{0}^{\pi}(1+\cos x) d x$$

You are sitting on the bank of a tidal river watching the incoming tide carry a bottle upstream. You record the velocity of the flow every 5 minutes for an hour, with the results shown in the accompanying table. About how far upstream did the bottle travel during that hour? Find an estimate using 12 sub intervals of length 5 with a. left-endpoint values. b. right-endpoint values. $$\begin{aligned} &\begin{array}{cc|cc} \hline \begin{array}{c} \text { Time } \\ (\text { min }) \end{array} & \begin{array}{c} \text { Velocity } \\ (\mathbf{m} / \mathrm{sec}) \end{array} & \begin{array}{c} \text { Time } \\ (\mathrm{min}) \end{array} & \begin{array}{c} \text { Velocity } \\ (\mathbf{m} / \mathrm{sec}) \end{array} \\ \hline 0 & 1 & 35 & 1.2 \\ 5 & 1.2 & 40 & 1.0 \\ 10 & 1.7 & 45 & 1.8 \\ 15 & 2.0 & 50 & 1.5 \\ 20 & 1.8 & 55 & 1.2 \\ 25 & 1.6 & 60 & 0 \\ 30 & 1.4 & & \\ \hline \end{array}\\\ \end{aligned}$$

Which formula is not equivalent to the other two? a. \(\sum_{k=1}^{4}(k-1)^{2}\) b. \(\sum_{k=-1}^{3}(k+1)^{2}\) c. \(\sum_{k=-3}^{-1} k^{2}\)

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. a. \(\int_{1}^{2} f(u) d u\) b. \(\int_{1}^{2} \sqrt{3} f(z) d z\) c. \(\int_{2}^{1} f(t) d t\) d. \(\int_{1}^{2}[-f(x)] d x\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \frac{9 r^{2} d r}{\sqrt{1-r^{3}}}, \quad u=1-r^{3}$$

You and a companion are about to drive a twisty stretch of dirt road in a car whose speedometer works but whose odometer (mileage counter) is broken. To find out how long this particular stretch of road is, you record the car's velocity at 10 -sec intervals, with the results shown in the accompanying table. Estimate the length of the road using a. left-endpoint values. b. right-endpoint values. $$\begin{aligned} &\begin{array}{cc|cc} \hline & \text { Velocity } & & \text { Velocity } \\ \text { Time } & \text { (converted to ft/sec) } & \text { Time } & \text { (converted to ft/sec) } \\ \text { (sec) } & (30 \mathrm{mi} / \mathrm{h}=44 \mathrm{ft} / \mathrm{sec}) & \text { (sec) } & (30 \mathrm{mi} / \mathrm{h}=44 \mathrm{ft} / \mathrm{sec}) \\ \hline 0 & 0 & 70 & 15 \\ 10 & 44 & 80 & 22 \\ 20 & 15 & 90 & 35 \\ 30 & 35 & 100 & 44 \\ 40 & 30 & 110 & 30 \\ 50 & 44 & 120 & 35 \\ 60 & 35 & & \\ \hline \end{array} \end{aligned}$$

Evaluate the integrals. $$\int_{\pi / 4}^{3 \pi / 4} \csc \theta \cot \theta d \theta$$

Express the sums in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$1+2+3+4+5+6$$

Suppose that \(\int_{-3}^{0} g(t) d t=\sqrt{2} .\) Find a. \(\int_{0}^{-3} g(t) d t\) b \(\int_{-3}^{0} g(u) d u\) c. \(\int_{-3}^{0}[-g(x)] d x\) d. \(\int_{-3}^{0} \frac{g(r)}{\sqrt{2}} d r\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int 12\left(y^{4}+4 y^{2}+1\right)^{2}\left(y^{3}+2 y\right) d y, \quad u=y^{4}+4 y^{2}+1$$

Evaluate the integrals. $$\int_{0}^{\pi / 3} 4 \frac{\sin u}{\cos ^{2} u} d u$$

Express the sums in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$1+4+9+16$$

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. \(\int_{4}^{3} f(t) d t\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \sqrt{x} \sin ^{2}\left(x^{3 / 2}-1\right) d x, \quad u=x^{3 / 2}-1$$

Free fall with air resistance An object is dropped straight down from a helicopter. The object falls faster and faster but its acceleration (rate of change of its velocity) decreases over time because of air resistance. The acceleration is measured in \(\mathrm{ft} / \mathrm{sec}^{2}\) and recorded every second after the drop for 5 sec, as shown: $$\begin{array}{c|cccccc} t & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline a & 32.00 & 19.41 & 11.77 & 7.14 & 4.33 & 2.63 \end{array}$$ a. Find an upper estimate for the speed when \(t=5.\) b. Find a lower estimate for the speed when \(t=5.\) c. Find an upper estimate for the distance fallen when \(t=3.\)

Evaluate the integrals. $$\int_{\pi / 2}^{0} \frac{1+\cos 2 t}{2} d t$$

Express the sums in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Suppose that \(h\) is integrable and that \(\int_{-1}^{1} h(r) d r=0\) and \(\int_{-1}^{3} h(r) d r=6 .\) Find A. \(\int_{1}^{3} h(r) d r\) B. \(-\int_{3}^{1} h(u) d u\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \frac{1}{x^{2}} \cos ^{2}\left(\frac{1}{x}\right) d x, \quad u=-\frac{1}{x}$$

An object is shot straight upward from sea level with an initial velocity of \(400 \mathrm{ft} / \mathrm{sec}.\) a. Assuming that gravity is the only force acting on the object, give an upper estimate for its velocity after 5 sec have elapsed. Use \(g=32 \mathrm{ft} / \mathrm{sec}^{2}\) for the gravitational acceleration. b. Find a lower estimate for the height attained after 5 sec.

Evaluate the integrals. $$\int_{-\pi / 3}^{\pi / 3} \sin ^{2} t d t$$

Express the sums in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$2+4+6+8+10$$

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). $$\int_{0}^{1} \sqrt{t^{5}+2 t}\left(5 t^{4}+2\right) d t$$

Graph the integrands and use known area formulas to evaluate the integrals. $$\int_{-2}^{4}\left(\frac{x}{2}+3\right) d x$$

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. \(\int \csc ^{2} 2 \theta \cot 2 \theta d \theta\) a. Using \(u=\cot 2 \theta\) b. Using \(u=\csc 2 \theta\)

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. $$f(x)=x^{3} \quad \text { on } \quad[0,2]$$ (Graph cant copy)

Evaluate the integrals. $$\int_{0}^{\pi / 4} \tan ^{2} x d x$$

Express the sums in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}$$

Graph the integrands and use known area formulas to evaluate the integrals. $$\int_{1 / 2}^{3 / 2}(-2 x+4) d x$$

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. \(\int \frac{d x}{\sqrt{5 x+8}}\) a. Using \(u=5 x+8\) b. Using \(u=\sqrt{5 x+8}\)

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)

Evaluate the integrals. $$\int_{0}^{\pi / 6}(\sec x+\tan x)^{2} d x$$

Express the sums in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$-\frac{1}{5}+\frac{2}{5}-\frac{3}{5}+\frac{4}{5}-\frac{5}{5}$$

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

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

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

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. $$f(t)=(1 / 2)+\sin ^{2} \pi t \quad \text { on } \quad[0,2]$$ (Graph cant copy)

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

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. \(\sum_{k=1}^{n} \frac{b_{k}}{6}\) c. \(\sum_{k=1}^{n}\left(a_{k}+b_{k}\right)\) d. \(\sum_{k=1}^{n}\left(a_{k}-b_{k}\right)\) e. \(\sum_{k=1}^{n}\left(b_{k}-2 a_{k}\right)\)

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). $$\int_{\pi}^{3 \pi / 2} \cot ^{5}\left(\frac{\theta}{6}\right) \sec ^{2}\left(\frac{\theta}{6}\right) d \theta$$

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

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

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. $$f(t)=1-\left(\cos \frac{\pi t}{4}\right)^{4} \text { on } [0,4]$$ (Graph cant copy)

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

Suppose that \(\sum_{k=1}^{n} a_{k}=0\) and \(\sum_{k=1}^{n} b_{k}=1 .\) Find the values of a. \(\sum_{k=1}^{n} 8 a_{k}\) 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 in Exercises \(1-46\). $$\int_{0}^{\pi} 5(5-4 \cos t)^{1 / 4} \sin t d t$$

Graph the integrands and use known area formulas to evaluate the integrals. $$\int_{-2}^{1}|x| d x$$

Evaluate the integrals. $$\int \theta \sqrt[4]{1-\theta^{2}} d \theta$$

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?

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