Chapter 22

For Problems 1 and 2, evaluate the following. (a) \(\int_{2}^{0} x d x\) (b) \(\int_{4}^{-1}(x+1) d x\)

By interpreting the de nite integral as signed area, calculate the following. (a) \(\int_{0}^{5} x d x\) (b) \(\int_{-2}^{2} x d x\) (c) \(\int_{-2}^{2}|x| d x\) (d) \(\int_{-1}^{3} 3 d x\) (e) \(\int_{0}^{2 \pi} \sin t d t\) (f) \(\int_{-\pi}^{\pi} \cos z d z\) (g) \(\int_{-2}^{2}(x+1) d x\) (h) \(\int_{-2}^{2}|x+1| d x\)

Summation notation review (a) Write the following in summation notation. i. \(3-4+5-6+7-\cdots-300\) ii. \(2+4+6+\cdots+1000\) iii. \(1+3+5+\cdots+999\) iv. \(\frac{2}{3}-\frac{2}{9}+\frac{2}{27}-\frac{2}{81}+\cdots+\frac{2}{3^{15}}\) v. \(x+x^{2}+x^{3}+x^{4}+\cdots+x^{40}\) vi. \(1^{2}+2^{2}+3^{2}+\cdots+100^{2}\) vii. \(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\cdots+a_{n} x^{n}\) (b) Write out the following sums. i. \(\sum_{i=2}^{5} i^{2}\) ii. \(\sum_{k=0}^{4} 2^{k}\) iii. \(\sum_{j=0}^{3} a_{j} x^{j}\)

Maple syrup is being poured at a decreasing rate out of a tank. By taking readings from the valve on the tank, we have the following information on the rate at which the syrup is leaving the tank.$$ \begin{array}{lccccc} t \text { (seconds) } & 0 & 2 & 4 & 6 & 8 \\ \left.\hline \text { rate (in cm }^{3} / \mathrm{sec}\right) & 10 & 9 & 7 & 4 & 2 \end{array} $$ (a) Find a good upper bound for the amount of maple syrup that has been poured out between time \(t=0\) and \(t=8\). (b) Find a good lower bound for this same amount.

evaluate the following. a) \(\int_{0}^{1} \sqrt{1-x^{2}} d x\) (b) \(\int_{1}^{-1} \sqrt{1-x^{2}} d x\)

Find the following and express your answer as simply as possible. (a) \(\sum_{k=1}^{10}\left(\frac{k}{5}\right)^{2}-\sum_{k=0}^{9}\left(\frac{k}{5}\right)^{2}\) (b) \(\sum_{k=1}^{n}\left(\frac{k}{n}\right)^{2}-\sum_{k=0}^{n-1}\left(\frac{k}{n}\right)^{2}\)

An industrial chemist is making a mixture in a large container. A certain chemical, B, is being introduced into the mixture at an ever-increasing rate. Some of the rates have been registered below. Time \(t=0\) marks the rst introduction of this chemical into the mixture. $$ \begin{array}{lcccccc} \text { time (in minutes) } & 0 & 3 & 7 & 10 & 13 & 15 \\ \hline \text { rate (in grams/min.) } & 10 & 12 & 20 & 23 & 25 & 29 \end{array} $$ Determine reasonable upper and lower bounds for the number of grams of chemical B in the mixture at time \(t=13\).

Choose the correct answer and explain your reasoning. (a) \(\int_{-a}^{a} \frac{1}{1+x^{2}} d x=\) (i) 0 (ii) \(\int_{0}^{a} \frac{1}{1+x^{2}} d x\) (iii) \(2 \int_{0}^{a} \frac{1}{1+x^{2}} d x\) (iv) \(-\int_{0}^{a} \frac{1}{1+x^{2}} d x\) (b) \(\int_{-a}^{a} \frac{x}{1+x^{2}} d x=\) (i) 0 (ii) \(\int_{0}^{a} \frac{x}{1+x^{2}} d x\) (iii) \(2 \int_{0}^{a} \frac{x}{1+x^{2}} d x\) (iv) \(-\int_{0}^{a} \frac{x}{1+x^{2}} d x\)

Put the following integrals in ascending order, placing \(<\) or \(=\) signs between them as appropriate. (Strategy: First determine which integrals are positive, which are negative, and which are zero.) (a) \(\int_{0}^{\pi} \sin (t) d t\) (b) \(\int_{-\pi}^{2 \pi} \sin (t) d t\) (c) \(\int_{-\pi}^{2 \pi} \cos (t) d t\) (d) \(\int_{0}^{\pi / 2} \cos (t) d t\) (e) \(\int_{0}^{\pi} \cos (2 t) d t\) (f) \(\int_{0}^{3 \pi / 2}|\sin (t)| d t\)

Find upper and lower bounds for each of the following de nite integrals by calculating left- and right-hand Riemann sums with the number of subdivisions indicated. (a) \(\int_{0}^{2} x^{3} d x \quad(n=4)\) (b) \(\int_{1}^{3} \frac{1}{t} d t \quad(n=6)\)

(a) By partitioning the interval \([0, \pi / 2]\) into four equal pieces and using the areas of inscribed and circumscribed rectangles as appropriate, nd upper and lower bounds for the area between the graph of \(\sin x\) and the \(x\) -axis for \(x\) in the interval \([0, \pi / 2] .\) Draw a picture illustrating what you have done. (Note: You will have to use your calculator to get some of the values of \(\sin x\) and to get a numerical answer.) (b) Using the work you did in part (a), nd upper and lower bounds for the area under the graph of \(\sin x\) between \(x=0\) and \(x=\pi\). Explain what you have done using a picture. (c) Using the work you did in part (b), give upper and lower bounds for the area under the graph of \(\cos x\) between \(x=-\pi / 2\) and \(\pi / 2\).

(a) Explain why \(\int_{0.5}^{3} \frac{1-\ln x}{x^{2}+1} d x>\int_{0.5}^{4} \frac{1-\ln x}{x^{2}+1} d x\). (Hint: Look at the sign of the integrand.) (b) Put in ascending order: \(0, \int_{1 / e}^{1} \frac{1-\ln x}{x^{2}+1} d x, \quad \int_{1 / e}^{2} \frac{1-\ln x}{x^{2}+1} d x, \quad \int_{1 / e}^{e} \frac{1-\ln x}{x^{2}+1} d x, \quad \int_{1 / e}^{4} \frac{1-\ln x}{x^{2}+1} d x\)

Suppose velocity (in miles per hour) is given by \(v(t)=3 t\), where \(t\) is measured in hours. We are interested in the distance traveled from \(t=0\) to \(t=k\), where \(k\) is a constant. (a) By solving the differential equation \(d s / d t=3 t\) and using the initial condition \(s(0)=s_{0}\), nd the distance function \(s(t) .\) Using \(s(t)\), nd i. \(s(0)\). ii. \(s(k)\). iii. the distance traveled between \(t=0\) and \(t=k\). (b) Find the area under the graph of \(v(t)\) from \(t=0\) to \(t=k\). Verify that your answers to part (a) iii and (b) are the same.

(a) Assume that \(a

Suppose we want to approximate \(\int_{0}^{2} f(x) d x\) by partitioning the interval \([0,2]\) into \(n\) equal pieces and constructing left- and right-hand sums. Let \(f(x)=x^{3}\). (a) Put the following expressions in ascending order. $$ \int_{0}^{2} f(x) d x, \quad L_{4}, \quad R_{4}, \quad L_{20}, \quad R_{20}, \quad L_{100}, \quad R_{100} $$ (b) Find \(\left|R_{4}-L_{4}\right|\). (c) Find \(\left|R_{100}-L_{100}\right|\). (d) How large must \(n\) be to assure that \(\left|R_{n}-L_{n}\right|<0.05\) ? (e) Write out \(R_{4}\), once using summation notation, once without. Evaluate \(R_{4}\).

Suppose velocity (in miles per hour) is given by \(v(t)=m t+\mathrm{c}\), where \(m\) and \(c\) are positive constants. (a) Using your knowledge of the area of a trapezoid, nd the area under the graph of \(v(t)\) on the interval \([a, b]\), where \(a\) and \(b\) are positive constants. (b) By solving the differential equation \(d s / d t=m t+c\) and using the initial condition \(s(0)=s_{0}\), nd the distance function \(s(t) .\) Using \(s(t)\) nd i. \(s(a)\). ii. \(s(b)\). iii. the distance traveled between \(t=a\) and \(t=b\). Verify that your answers to parts (a) and (b) iii are the same.

For each of the following, sketch a graph of the indicated region and write a de nite integral (or, if you prefer, the sum and/or differences of de nite integrals) that gives the area of the region. (a) The area between the horizontal line \(y=4\) and the parabola \(y=x^{2}\) (b) The area between the line \(y=x+1\) and the parabola \(y=x^{2}-1\)

Put the following four integrals in ascending order (from smallest to largest). Explain, using graphs, how you can be sure that the order you gave is correct. $$ \int_{0}^{\pi} \sin t d t, \quad \int_{0}^{\pi} 2 \sin t d t, \quad \int_{0}^{\pi} \sin (2 t) d t, \quad \int_{\pi}^{0} \sin t d t $$

(a) What is the equation of a circle of radius 2 centered at the origin? (b) Write a function, complete with domain, that gives the equation of the top half of a circle of radius 2 centered at the origin. (c) Let \(f(x)=\left\\{\begin{array}{ll}\sqrt{4-x^{2}} & \text { for }-2 \leq x \leq 0 \text { , } \\ 2 x & \text { for } x>0\end{array}\right.\) Evaluate \(\int_{-2}^{2} f(x) d x\)

(a) The velocity of an object at time \(t\) is given by \(v(t)=t^{2} \mathrm{ft} / \mathrm{sec} .\) Partition the time interval \([0,3]\) into 3 equal pieces each of length 1 second. Find upper and lower bounds for the distance the object traveled between time \(t=0\) and \(t=3\). (b) Illustrate your work in part (a) by graphing \(v(t)\) and using areas of inscribed and circumscribed rectangles. Draw two pictures, one illustrating the upper bound and the other the lower bound. (c) Repeat part (a), but this time partition the interval into 6 equal pieces, each of length \(1 / 2\). Make a sketch indicating the areas you have found. (d) What is the difference between \(R_{n}\) and \(L_{n}\) if the interval is partitioned into 50 equal pieces? 100 equal pieces? (e) Into how many equal pieces must we partition \([0,3]\) to be sure that the difference between the right- and left-hand sums is less than or equal to \(0.01\) ?

Suppose \(v(t)\) gives the velocity of a trekker on the time interval \([0,3]\) and suppose that \(v(t)\) is positive and decreasing over this interval. If we use a left-hand sum to approximate the distance she has covered over this time interval, will the approximation give a lower bound or an upper bound?

If \(\int_{0}^{5} f(t) d t=10\) and \(\int_{0}^{5} g(t) d t=3\) does it follow that (a) \(f(t)>g(t)\) for all \(t\) between 0 and \(5 ?\) Explain. Mathematicians might write this statement in mathematical symbols as follows: \(f(t)>g(t) \forall t\) in \([0,5]\). The symbol \(\forall\) is read for all. (b) \(f(t)>g(t)\) for some \(t\) between 0 and 5 ? Explain. Mathematicians might write this statement in mathematical symbols as follows: \(\exists t\) in \([0,5]\) such that \(f(t)>g(t)\). The symbol \(\exists\) is read there exists.