적분법 전체 문제

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.

L₄ for $\frac{1}{{x}^{2}+1}$ 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.

R₈ for ${x}^{2}-2x+1$ 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.

L₈ for ${x}^{2}-2x+1$ on $[0,2]$

Compute the left and right Riemann sums—L₄ and R₄, 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—L₆ and R₆, 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—L₄ and R₄, respectively—for $f(x)=\sqrt{4-{x}^{2}}$ on $[-2,2]$ and compare their values.

Express the following endpoint sums in sigma notation but do not evaluate them.

L₃₀ for $f(x)={x}^{2}$ on $[1,2]$

Express the following endpoint sums in sigma notation but do not evaluate them.

L₁₀ for $f(x)=\sqrt{4-{x}^{2}}$ on $[-2,2]$

Express the following endpoint sums in sigma notation but do not evaluate them.

R₂₀ for $f(x)=\sin x$ on $[0,\pi ]$

Express the following endpoint sums in sigma notation but do not evaluate them.

R₁₀₀ for $\ln x$ on $[1,e]$

[T] L₁₀₀ and R₁₀₀ for $y={x}^{2}-x+3$ on the interval $[-1,1]$

[T] L₅₀ and R₅₀ for $y=\frac{x+1}{{x}^{2}-1}$ on the interval $[2,4]$

[T] L₁₀₀ and R₁₀₀ for $y={x}^{3}+2$ on the interval $[-1,1]$

[T] L₅₀ and R₅₀ for $y=\tan (x)$ on the interval $\left[0,\frac{\pi }{4}\right]$

[T] L₁₀₀ and R₁₀₀ for $y={e}^{2x}$ on the interval $[-1,1]$

Let t_j 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 $\frac{1}{10}$ 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 (CO₂) each decade since 1960, in parts per million (ppm). Estimate the total increase in atmospheric CO₂ between 1964 and 2013.

Average Annual Atmospheric CO₂ Increase, 1964–2013

Source: http://www.esrl.noaa.gov/gmd/ccgg/trends/.