적분법 전체 문제

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.

Approximate 20-Year Sea Level Increases, 1870–1990

Source: http://link.springer.com/article/10.1007%2Fs10712-011-9119-1

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?

Approximate 10-Year Gas Price Increases, 1950–2000

Source: http://epb.lbl.gov/homepages/Rick_Diamond/docs/lbnl55011-trends.pdf.

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.

Annual Percentage Growth of U.S. Population, 2000–2009

Source: http://www.census.gov/popest/data.

(Hint: To obtain the population in July 2001, multiply the population in July 2000 by 1.0112 to get 284,573,831.)

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L₈.

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L₈.

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L₈.

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L₈.

[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 $[3,5]$

[T] $y={x}^{4}-5{x}^{2}+4$ on the interval $[-2,2],$ which has an exact area of $\frac{32}{15}$

[T] $y=\ln x$ on the interval $[1,2],$ which has an exact area of $2\ln (2)-1$

Explain why, if $f(a)\ge 0$ 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 $[a,b].$

Explain why, if $f(b)\ge 0$ and f is decreasing on $[a,b],$ that the left endpoint estimate is an upper bound for the area below the graph of f on $[a,b].$

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums R_N and L_N for $N=1,10,100.$

Show that, in general, ${R}_{N}-{L}_{N}=(b-a)\,\times \,\frac{f(b)-f(a)}{N}.$

Explain why, if f is increasing on $[a,b],$ the error between either L_N or R_N and the area A below the graph of f is at most $(b-a)\frac{f(b)-f(a)}{N}.$

For each of the three graphs:

a. Obtain a lower bound $L(A)$ for the area enclosed by the curve by adding the areas of the squares enclosed completely by the curve. b. Obtain an upper bound $U(A)$ for the area by adding to $L(A)$ the areas $B(A)$ of the squares enclosed partially by the curve.

In the previous exercise, explain why $L(A)$ gets no smaller while $U(A)$ gets no larger as the squares are subdivided into four boxes of equal area.