Chapter 5

State whether the given sums are equal or unequal. $$a\sum_{i=1}^{10} i and \sum_{k=1}^{10} k$$ $$b. \sum_{i=1}^{10} i and \sum_{i=6}^{15}(i-5)$$ $$c. \sum_{i=1}^{10} i(i-1) and \sum_{j=0}^{9}(j+1) j$$ $$d. \sum_{i=1}^{10} i(i-1) \text { and } \sum_{k=1}^{10}\left(k^{2}-k\right)$$

In the following exercises, use the rules for sums of powers of integers to compute the sums. \(\sum_{i=5}^{10} i\)

In the following exercises, use the rules for sums of powers of integers to compute the sums. $$\sum_{i=5}^{10} i^{2}$$

Suppose that \(\sum_{i=1}^{100} a_{i}=15\) and \(\sum_{i=1}^{100} b_{i}=-12 .\) In the following exercises, compute the sums. $$\sum_{i=1}^{100}\left(a_{i}+b_{i}\right) $$

Suppose that \(\sum_{i=1}^{100} a_{i}=15\) and \(\sum_{i=1}^{100} b_{i}=-12 .\) In the following exercises, compute the sums. $$\sum_{i=1}^{100}\left(a_{i}-b_{i}\right)$$

Suppose that \(\sum_{i=1}^{100} a_{i}=15\) and \(\sum_{i=1}^{100} b_{i}=-12 .\) In the following exercises, compute the sums. $$\sum_{i=1}^{100}\left(3 a_{i}-4 b_{i}\right)$$

Suppose that \(\sum_{i=1}^{100} a_{i}=15\) and \(\sum_{i=1}^{100} b_{i}=-12 .\) In the following exercises, compute the sums. $$\sum_{i=1}^{100}\left(5 a_{i}+4 b_{i}\right)$$

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. $$\sum_{k=1}^{20} 100\left(k^{2}-5 k+1\right)$$

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. $$\sum_{j=1}^{50}\left(j^{2}-2 j\right)$$

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. $$\sum_{j=11}^{20}\left(j^{2}-10 j\right)$$

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. $$\sum_{k=1}^{25}\left[(2 k)^{2}-100 k\right]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$L_{4} \text { for } f(x)=\frac{1}{x-1} \text { on }[2,3]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$R_{4} \text { for } g(x)=\cos (\pi x) \text { on }[0,1]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$L_{6} \text { for } f(x)=\frac{1}{x(x-1)} \text { on }[2,5]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$R_{6} \text { for } f(x)=\frac{1}{x(x-1)} \text { on }[2,5]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$L_{4} \text { for } \frac{1}{x^{2}+1} \text { on }[-2,2]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$R_{4} \text { for } x^{2}-2 x+1 \text { on }[0,2]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$L_{8} \text { for } x^{2}-2 x+1 \text { on }[0,2]$$

Compute the left and right Riemann sums \(-L_{4}\) and \(R_{4}\) , respectively- for \(f(x)=(2-|x|)\) on \([-2,2] .\) Compute their average value and compare it with the area under the graph of \(f .\)

Compute the left and right Riemann sums \(-L_{4}\) and \(R_{4}\) respectively - for \(\quad f(x)=(3-|3-x|) \quad\) on \(\quad[0,6]\) Compute their average value and compare it with the area under the graph of f.

Compute the left and right Riemann sums- \(L_{6}\) and \(R_{6},\) respectively-for \(f(x)=(3-|3-x|)\) on [0,6] Compute their average value and compare it with the area under the graph of \(f\).

Compute the left and right Riemann sums \(-L_{4}\) and \(R_{4}\) respectively - for \(f(x)=\sqrt[1]{4-x^{2}}\) on [?2, 2] and compare their values.

Compute the left and right Riemann sums \(-L_{6}\) and \(R_{6}\) respectively - for \(f(x)=\sqrt{9-(x-3)^{2}}\) on \([0,6]\) and compare their values.

Express the following endpoint sums in sigma notation but do not evaluate them. $$L_{30} \text { for } f(x)=x^{2} \text { on }[1,2]$$

Express the following endpoint sums in sigma notation but do not evaluate them. $$L_{10} \text { for } f(x)=\sqrt{4-x^{2}} \text { on }[-2,2]$$

Express the following endpoint sums in sigma notation but do not evaluate them. $$R_{20} \text { for } f(x)=\sin x \text { on }[0, \pi]$$

Express the following endpoint sums in sigma notation but do not evaluate them. \(R_{100}\) for \(\ln x\) on \([1, e]\)

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? [T] \(L_{100}\) and \(R_{100}\) for \(y=x^{2}-3 x+1\) on the interval \([-1,1]\)

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? \([\mathrm{T}] L_{100}\) and \(R_{100}\) for \(y=x^{2}\) on the interval \([0,1]\)

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? \([\mathrm{T}] L_{50}\) and \(R_{50}\) for \(y=\frac{x+1}{x^{2}-1}\) on the interval \([2,4]\)

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? \([\mathrm{T}] L_{100}\) and \(R_{100}\) for \(y=x^{3}\) on the interval \([-1,1]\)

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? \([\mathrm{T}] L_{50}\) and \(R_{50}\) for \(y=\tan (x)\) on the interval \(\left[0, \frac{\pi}{4}\right]\)

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? \([\mathrm{T}] L_{100}\) and \(R_{100}\) for \(y=e^{2 x}\) on the interval \([-1,1]\)

Let tj denote the time that it took Tejay van Garteren to ride the jth stage of the Tour de France in 2014. If there were a total of 21 stages, interpret \(\sum_{j=1}^{21} t_{j}\)

Let \(r_{j}\) denote the total rainfall in Portland on the jth day of the year in 2009 . Interpret \(\sum_{j=1}^{31} r_{j}\)

Let \(d_{j}\) denote the hours of daylight and \(\delta_{j}\) denote the increase in the hours of daylight from day \(j-1\) to day \(j\) in Fargo, North Dakota, on the jth day of the year. Interpret \(d_{1}+\sum_{j=2}^{365} \delta_{j}\)

To help get in shape, Joe gets a new pair of running shoes. If Joe runs 1 mi each day in week 1 and adds \(\frac{1}{10} \mathrm{mi}\) to his daily routine each week, what is the total mileage on Joe's shoes after 25 weeks?

The following table gives approximate values of the average annual atmospheric rate of increase in carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) each decade since 1960 , in parts per million (ppm). Estimate the total increase in atmospheric \(\mathrm{CO}_{2}\) between 1964 and 2013 . $$ \begin{array}{|c|c|} \hline \text { Decade } & \text { Ppm/y } \\ \hline 1964-1973 & 1.07 \\ \hline 1974-1983 & 1.34 \\ \hline 1984-1993 & 1.40 \\ \hline 1994-2003 & 1.87 \\ \hline 2004-2013 & 2.07 \\ \hline \end{array} $$

The following table gives the approximate increase in sea level in inches over 20 years starting in the given year. Estimate the net change in mean sea level from 1870 to \(2010 .\) $$ \begin{array}{|c|l|} \hline \text { Starting Year } & \text { 20-Year Change } \\ \hline 1870 & 0.3 \\ \hline 1890 & 1.5 \\ \hline 1910 & 0.2 \\ \hline 1930 & 2.8 \\ \hline 1950 & 0.7 \\ \hline 1970 & 1.1 \\ \hline 1990 & 1.5 \\ \hline \end{array} $$

The following table gives the approximate increase in dollars in the average price of a gallon of gas per decade since 1950 . If the average price of a gallon of gas in 2010 was \(\$ 2.60\), what was the average price of a gallon of gas in \(1950 ?\) $$ \begin{array}{|c|c|} \hline \text { Starting Year } & \text { 10-Year Change } \\ \hline 1950 & 0.03 \\ \hline 1960 & 0.05 \\ \hline 1970 & 0.86 \\ \hline 1980 & -0.03 \\ \hline 1990 & 0.29 \\ \hline 2000 & 1.12 \\ \hline \end{array} $$

The following table gives the percent growth of the U.S. population beginning in July of the year indicated. If the U.S. population was 281,421,906 in July 2000, estimate the U.S. population in July 2010.

[IT] Use a computer algebra system to compute the Riemann sum, \(L_{N}, \quad\) for \(\quad N=10,30,50\) for \(f(x)=\sqrt{1-x^{2}}\) on \([-1,1]\)

[T] Use a computer algebra system to compute the Riemann sum, \(L_{N}, \quad\) for \(\quad N=10,30,50\) for \(f(x)=\sqrt{1-x^{2}}\) on [-1,1]

[T] Use a computer algebra system to compute the Riemann sum, \(L_{N}, \quad\) for \(\quad N=10,30,50\) for \(f(x)=\frac{1}{\sqrt{1+x^{2}}}\) on \([-1,1]\)

[T] Use a computer algebra system to compute the Riemann sum, \(L_{N},\) for \(N=10,30,50\) for \(f(x)=\sin ^{2} x\) on \([0,2 \pi] .\) Compare these estimates with \(\pi .\)

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums \(R_{N}\) and \(L_{N}\) for \(N=1,10,100\) . How do these estimates compare with the exact answers, which you can find via geometry? [T]\(y=\cos (\pi x)\) on the interval \([0,1]\)

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums \(R_{N}\) and \(L_{N}\) for \(N=1,10,100\) . How do these estimates compare with the exact answers, which you can find via geometry? \([T] \mathrm{y}=3 x+2\) on the interval \([3,5]\)

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums \(R_{N}\) and \(L_{N}\) for \(N=1,10,100\) . [T] \(y=x^{4}-5 x^{2}+4\) on the interval \([-2,2]\) which has an exact area of \(\frac{32}{15}\)

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums \(R_{N}\) and \(L_{N}\) for \(N=1,10,100\) . [T] \(y=\ln x\) on the interval \([1,2],\) which has an exact area of \(2 \ln (2)-1\)

Explain why, if \(f(a) \geq 0\) and \(f\) is increasing on \([a, b],\) that the left endpoint estimate is a lower bound for the area below the graph of \(f\) on \([a, b]\)