정적분과 급수(구분구적법)

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.

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/.

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.

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 $[a,b].$

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 Exercises 29–34, a definite integral

$\displaystyle \int_a^b f(x) \ dx$ is given.

(a) Graph $f(x)$ on $[a,b]$. (b) Add to the sketch rectangles using the provided rule. (c) Approximate $\displaystyle \int_a^b f(x) \ dx$ by summing the areas of the rectangles.

$\displaystyle \int_{-3}^3 x^2\ dx$, with 6 rectangles using the Left Hand Rule.

In Exercises 29–34, a definite integral

$\displaystyle \int_a^b f(x) \ dx$ is given.

(a) Graph $f(x)$ on $[a,b]$. (b) Add to the sketch rectangles using the provided rule. (c) Approximate $\displaystyle \int_a^b f(x) \ dx$ by summing the areas of the rectangles.

$\displaystyle \int_{0}^2 (5-x^2)\ dx$, with 4 rectangles using the Midpoint Rule.

In Exercises 29–34, a definite integral

$\displaystyle \int_a^b f(x) \ dx$ is given.

(a) Graph $f(x)$ on $[a,b]$. (b) Add to the sketch rectangles using the provided rule. (c) Approximate $\displaystyle \int_a^b f(x) \ dx$ by summing the areas of the rectangles.

$\displaystyle \int_{0}^\pi \sin x\ dx$, with 6 rectangles using the Right Hand Rule.

In Exercises 29–34, a definite integral

$\displaystyle \int_a^b f(x) \ dx$ is given.

(a) Graph $f(x)$ on $[a,b]$. (b) Add to the sketch rectangles using the provided rule. (c) Approximate $\displaystyle \int_a^b f(x) \ dx$ by summing the areas of the rectangles.

$\displaystyle \int_{0}^3 2^x\ dx$, with 5 rectangles using the Left Hand Rule.

In Exercises 29–34, a definite integral

$\displaystyle \int_a^b f(x) \ dx$ is given.

(a) Graph $f(x)$ on $[a,b]$. (b) Add to the sketch rectangles using the provided rule. (c) Approximate $\displaystyle \int_a^b f(x) \ dx$ by summing the areas of the rectangles.

$\displaystyle \int_{1}^2 \ln x\ dx$, with 3 rectangles using the Midpoint Rule.

In Exercises 29–34, a definite integral

$\displaystyle \int_a^b f(x) \ dx$ is given.

(a) Graph $f(x)$ on $[a,b]$. (b) Add to the sketch rectangles using the provided rule. (c) Approximate $\displaystyle \int_a^b f(x) \ dx$ by summing the areas of the rectangles.

$\displaystyle \int_{1}^9 \frac1x\ dx$, with 4 rectangles using the Right Hand Rule.