정적분과 급수(구분구적법)

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 $f(x)=\sqrt{4-{x}^{2}}$ 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 $\sum_{j=1}^{31}{r}_{j}.$

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?

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

[T] Use a computer algebra system to compute the Riemann sum, L_N, for $N=10,30,50$ 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 π.

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] $y=3x+2$ on the interval $[3,5]$