5.1 Approximating areas  (Page 8/17)

L ₄ for $f\left(x\right)=\frac{1}{x-1}$ on $\left[2,3\right]$

R ₄ for $g\left(x\right)=\text{cos}\left(\pi x\right)$ on $\left[0,1\right]$

L ₆ for $f\left(x\right)=\frac{1}{x\left(x-1\right)}$ on $\left[2,5\right]$

R ₆ for $f\left(x\right)=\frac{1}{x\left(x-1\right)}$ on $\left[2,5\right]$

R ₄ for $\frac{1}{x^2+1}$ on $\left[\mathrm{-2},2\right]$

L ₄ for $\frac{1}{x^2+1}$ on $\left[\mathrm{-2},2\right]$

R ₄ for $x^2-2x+1$ on $\left[0,2\right]$

L ₈ for $x^2-2x+1$ on $\left[0,2\right]$

Compute the left and right Riemann sums— L ₄ and R ₄ , respectively—for $f\left(x\right)=\left(2-\left|x\right|\right)$ on $\left[\mathrm{-2},2\right].$ 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 $f\left(x\right)=\left(3-\left|3-x\right|\right)$ on $\left[0,6\right].$ 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 $f\left(x\right)=\sqrt{4-x^2}$ on $\left[\mathrm{-2},2\right]$ and compare their values.

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

L ₃₀ for $f\left(x\right)=x^2$ on $\left[1,2\right]$

L ₁₀ for $f\left(x\right)=\sqrt{4-x^2}$ on $\left[\mathrm{-2},2\right]$

R ₂₀ for $f\left(x\right)=\text{sin}x$ on $\left[0,\pi \right]$

R ₁₀₀ for $\text{ln}x$ on $\left[1,e\right]$

[T] L ₁₀₀ and R ₁₀₀ for $y=x^2-3x+1$ on the interval $\left[\mathrm{-1},1\right]$

[T] L ₁₀₀ and R ₁₀₀ for $y=x^2$ on the interval $\left[0,1\right]$

[T] L ₅₀ and R ₅₀ for $y=\frac{x+1}{x^2-1}$ on the interval $\left[2,4\right]$

[T] L ₁₀₀ and R ₁₀₀ for $y=x^3$ on the interval $\left[\mathrm{-1},1\right]$

[T] L ₅₀ and R ₅₀ for $y=\text{tan}\left(x\right)$ on the interval $\left[0,\frac{\pi }{4}\right]$

[T] L ₁₀₀ and R ₁₀₀ for $y=e^{2x}$ on the interval $\left[\mathrm{-1},1\right]$

Let t _j denote the time that it took Tejay van Garteren to ride the j th stage of the Tour de France in 2014. If there were a total of 21 stages, interpret ${\displaystyle \sum _{j=1}^{21}{t}_j}.$

Let $r_j$ denote the total rainfall in Portland on the j th day of the year in 2009. Interpret ${\displaystyle \sum _{j=1}^{31}{r}_j}.$

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 j th day of the year. Interpret $d_1+{\displaystyle \sum _{j=2}^{365}{\delta }_j}.$

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?