Chapter 9

Find the generic terms for the following sums, and write the sums using \(\Sigma\) notation. a. \(2^{2}+4^{2}+6^{2}+8^{2}+10^{2}\) b. \(\quad \frac{1}{1.1}+\frac{1}{1.2}+\frac{1}{1.3}+\frac{1}{1.4}+\frac{1}{1.5}+\frac{1}{1.6}+\frac{1}{1.7}+\frac{1}{1.8}+\frac{1}{1.9}+\frac{1}{2}\) c. \(\left[\left(\frac{1}{1}\right)^{2}+\left(\frac{1}{1.1}\right)^{2}+\left(\frac{1}{1.2}\right)^{2}+\cdots+\left(\frac{1}{1.8}\right)^{2}+\left(\frac{1}{1.9}\right)^{2}\right] \times \frac{1}{10}\) d. \(\quad\left[(-1.0)^{3}+(-0.9)^{3}+(-0.8)^{3}++\cdots+(-0.2)^{3}+(-0.1)^{3}\right] \times \frac{1}{10}\) e. \(\left[\sqrt{1-0.1^{2}}+\sqrt{1-0.2^{2}}+\sqrt{1-0.3^{2}}+\cdots+\sqrt{1-0.9^{2}}+\sqrt{1-1.0^{2}}\right] \times \frac{1}{10}\) f. \([\sqrt{0.1}+\sqrt{0.2}+\sqrt{0.3}+\cdots+\sqrt{1.9}+\sqrt{2.0}] \times \frac{1}{10}\)

Is the exponential function, \(E(x)=e^{x}\), linear? Prove or disprove.

Evaluate: a. \(\sum_{k=1}^{100} k\) b. \(\sum_{k=3}^{100} k\) c. \(\sum_{k=18}^{100} k \quad\) d. \(\quad \sum_{k=1}^{100} 2 k\) e. \(\sum_{k=1}^{10} k^{2}\) f. \(\sum_{k=3}^{100} k^{3}\) g. \(\quad \sum_{k=1}^{100}(k+5)\) h. \(\sum_{k=50}^{100}(3 k+7)\) i. \(\quad \sum_{k=1}^{10} 5 k^{2} \quad\) j. \(\quad \sum_{k=3}^{100}\left(k^{3}-9\right)\) k. \(\sum_{k=32}^{78}\left(3 k^{2}-k\right) \quad\) l. \(\quad\left(\sum_{k=50}^{100} 3 k\right)+7\)

Is the logarithm function, \(L(x)=\ln (x)\), linear? Prove or disprove.

Approximate A. \(\int_{0}^{1} e^{t} d t\) B. \(\int_{0}^{\pi} \sin (t) d t\)

Compute (note: change \(x\) to \(t\) if it confuses you.) a. \(\int_{0}^{1}\left[3+x^{2}\right] d x\) b. \(\int_{1}^{2} 3 x^{2} d x\) c. \(\int_{3}^{5} 3 x^{3}-6 x^{2} d x\)

Use Definition of Integral II to evaluate $$\int_{1}^{2} \frac{1}{t^{2}} d t$$ Partition [1,2] in \(n\) equal subintervals by $$t_{0}=1, \quad t_{1}=1+\frac{1}{n}, \quad \cdots, \quad t_{k-1}=1+\frac{k-1}{n}, \quad t_{k}=1+\frac{k}{n},\quad\cdots \quad t_{n}=1+\frac{n}{n}$$ Let $$\tau_{k}=\sqrt{t_{k-1} \times t_{k}}, \quad k=1,2, \cdots, n$$ a. Show that \(t_{k-1} \leq \tau_{k} \leq t_{k}\). b. Write Equation 9.16 , $$\int_{a}^{b} f(t) d t=\lim _{\|\Delta\| \rightarrow 0} \sum_{k=1}^{n} f\left(\tau_{k}\right) \times\left(t_{k}-t_{k-1}\right),$$ for \(\int_{1}^{2} \frac{1}{t^{2}} d t,\) the given partition and values of \(\tau_{k}\). c. Show that $$\int_{1}^{2} \frac{1}{t^{2}} d t=\lim _{\|\Delta\| \rightarrow 0} \sum_{k=1}^{n}\left(\frac{1}{t_{k-1}}-\frac{1}{t_{k}}\right)$$ d. Write the previous sum in long form and show that $$\int_{1}^{2} \frac{1}{t^{2}} d t=\lim _{\|\Delta\| \rightarrow 0}\left(1-\frac{1}{2}\right)=\frac{1}{2}$$

a. Compute: \(\mathrm{P}_{1} \quad \int_{2}^{4}\left[t \times t^{2}\right] d t\) and $$\mathrm{P}_{2} \int_{2}^{4} t d t \times \int_{2}^{4} t^{2} d t$$ b. Compute: \(\mathrm{Q}_{1}\) \(\int_{2}^{4} \frac{t^{3}}{t^{2}} d t \quad\) and \(\quad \mathrm{Q}_{2}\) $$\frac{\int_{2}^{4} t^{3} d t}{\int_{2}^{4} t^{2} d t}$$ c. What do these two problems illustrate?

Suppose \(f,\) is a continuous defined on an interval \([a, b]\) and \((v, f(v))\) is a high point of \(f\) on \([a, b]\) (meaning that \(v\) is in \([a, b]\) and for all \(x\) in \([a, b], f(x) \leq f(v))\). a. Argue that every approximating sum for \(\int_{a}^{b} f(x) d x\) is less than or equal to \(f(v) \times(b-a)\). b. Argue that$$\int_{a}^{b} f(x) d x \leq f(v) \times(b-a)$$ c. Argue that if \(\left(u, f(u)\right.\) is a low point of \(f\) on \([a, b],\) then \(f(u) \times(b-a) \leq \int_{a}^{b} f(x) d x\).

The exact area of the region bounded by the graphs of $$y=\sqrt{t} \quad y=0 \quad \text { and } \quad t=5$$ is \(\frac{2}{3} 5^{3 / 2} \doteq 7.453560\). Compute the upper rectangular approximation and the trapezoidal approximation to the area based on 50 subintervals and compute their relative accuracies.

Compute the area of the region bounded by the graph of \(y=t^{2},\) the \(t\) -axis, and the lines \(t=1\) and \(t=2\).

The exact area of the region bounded by the graphs of $$y=\sqrt{t} \quad y=0 \quad \text { and } \quad t=5 $$ is \(\frac{2}{3} 5^{3 / 2} \doteq 7.453560\). Compute the upper rectangular approximation and the trapezoidal approximation to the area based on 50 subintervals and compute their relative accuracies.

Write an integral that is the area of the region bounded by the graphs of a. \(y=t^{2}-t \quad\) and \(\quad y=0, \quad t=1, \quad t=2\). b. \(y=t^{2}, \quad\) and \(\quad y=t, \quad t=1, \quad t=2\). c. \(y=t^{2}, \quad\) and \(\quad y=t, \quad t=0, \quad t=1\). d. \(y=2 \times t^{5}-t^{4},\) and \(\quad y=0, \quad t=1, \quad t=2\). e. \(y=2 \times t^{5}, \quad\) and \(\quad y=t^{4}, \quad t=1, \quad t=2\). It is useful to sketch the regions.

If an object moves at a constant speed, \(s \mathrm{~m} / \mathrm{min}\), over a time interval \([a, b]\) minutes, the distance, \(D,\) traveled is \(D=s \mathrm{~m} / \mathrm{min} \times(b-a) \min , D=s \times(b-a)\) meters. If the speed \(s(t)\) varies over \([a, b]\), the distance, \(D,\) can be approximated by partitioning \([a, b]\) by \(a=t_{0}

Suppose a particle moves with a velocity, \(v(t)=\frac{1}{1+t^{2}}\) a. Write an integral that is the distance moved by the particle between times \(t=0\) and \(t=1\). b. Write an integral that is the distance moved by the particle between times \(t=-1\) and \(t=1\).

A consequence of some previous problems is: If \(n\) is either \(0,1,2,\) or \(3, x\) is a positive number, and \(A\) is the area of the region bounded by the graphs of $$y=t^{n} \quad y=0 \quad t=0 \quad \text { and } \quad t=x$$ then Case: \(\quad n=0 \quad A=x\) Case : \(\quad n=1 \quad A=\frac{x^{2}}{2}\) Case : \(\quad n=2 \quad A=\frac{x^{3}}{3} \quad\) Case : \(\quad n=3 \quad A=\frac{x^{4}}{4}\) Make a guess as to the values of \(A\) for \(n=4, n=10,\) and \(n=1,568\).

Suppose an item is drawn from a normal distribution that has mean 0 and standard deviation \(1\left(p(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}\right) .\) Write an integral for the probability that the item is a. less than one standard deviation from the mean. b. less than two standard deviations from the mean.

Suppose water is flowing into a barrel at the rate of \(R(t)=1+t^{2} \mathrm{~m}^{3} / \mathrm{min}\) for \(0 \leq t \leq 3\) minutes. Write an integral that is the volume of water put into the tank. Confirm that the units on the integral are volume.

A tank has \(30 \mathrm{~m}^{3}\) capacity and has \(15 \mathrm{~m}^{3}\) of water in it. Water flows into it at the rate of \(R(t)=1+t^{2} \mathrm{~m}^{3} / \mathrm{min}\) for \(0 \leq t \leq 3\) minutes. Does the tank overflow? $$\begin{aligned}&\text { Table } 9.2: P V \text { and } P \cdot(V+3.6) \text { for the data in Figure } 9.12 \text { . }\\\&\begin{array}{lllllllllll} P V & 608 & 606 & 595 & 587 & 577 & 567 & 558 & 544 & 538 & 519 \\ P(V+3.6) & 644 & 645 & 637 & 634 & 629 & 625 & 625 & 623 & 635 & 643 \end{array}\end{aligned}$$

Water flows into a tank at the rate of \(R(t)=1+t^{2} / 5 \mathrm{~m}^{3} / \mathrm{min}\) for \(0 \leq t \leq 3\) minutes and the concentration of salt in the water is \(C(t)=3.5 e^{-t} \mathrm{~g} / 1\) at time \(t\). Write an integral that is the total amount of salt that flowed into the tank. Confirm that the units on the integral are grams of salt.

Work against the force of gravity. Suppose a 3 kilogram mass instrument is lifted 20,000 kilometers above the surface of the Earth, the force acting on it is not constant throughout the motion. See Figure 9.14. The acceleration of gravity may be computed for a distance \(x\) above the Earth as Acceleration of gravity at altitude \(x=9.8 \times \frac{R^{2}}{(R+x)^{2}}\) where \(R \doteq 6,370\) kilometers is the radius of the Earth. We partition the interval [0,20,000] into 4 equal subintervals and assume the acceleration of gravity to be constant on each of the subintervals. The work to lift the instrument the first 5000 kilometers is approximately $$ 5.05 \times 3 \times 5000=75814 \quad \text { Newton-meters } $$ a. Approximate the total work to lift the instrument to \(20,000 \mathrm{~km}\). b. Approximate the total work done to lift the \(3 \mathrm{~kg}\) instrument \(40,000 \mathrm{~km}\). (You already know the work required to lift it \(20,000 \mathrm{~km} .)\) c. Approximate the total work done to lift the \(3 \mathrm{~kg}\) instrument \(100,000 \mathrm{~km}\). It is a very interesting question as to whether the instrument can 'escape from the Earth's gravity field' with a finite amount of work. We will return to the question in Section 11.5.1, Escape Velocity.

What is the norm of the partition \\{0.0,0.2,0.3,0.6,0.7,0.9,1.0\\} of [0,1]\(?\) Write a partition of [0,1] whose norm is 0.15 .

Write an approximating sum to the given integrals for the partition \\{0.0,0.2,0.3,0.6,0.7,0.9,1.0\\} of [0,1] a. \(\int_{0}^{1} 1 d x\) $$\begin{array}{l}\text { d. } \int_{0}^{1} x^{2} d x \\ \text { d. } \int_{0}^{1} \sin (\pi x) d x\end{array}$$ c. \(\int_{0}^{1} e^{x} d x\)

Exercise 9.4 .21 Let \(x\) be a number in \(\left[0, \frac{\pi}{2}\right]\). Use the trigonometric identity $$\sum_{k=1}^{n} \cos (k \times \theta)=\frac{\sin \left(n \times \theta+\frac{\theta}{2}\right)-\sin \left(\frac{\theta}{2}\right)}{2 \sin \left(\frac{\theta}{2}\right)}$$ and $$\lim _{h \rightarrow 0} \frac{\sin (h)}{h}=1$$ to compute from Definition I the integral $$\int_{0}^{x} \cos (t) d t=\sin x$$