left-endpoint approximation

an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle

a sum obtained by using the minimum value of $f (x)$ on each subinterval

a set of points that divides an interval into subintervals

a partition in which the subintervals all have the same width

an estimate of the area under the curve of the form $A \approx \sum_{i = 1}^{n} f (x_i^*) \text{Δ} x$

right-endpoint approximation

the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle

(also, summation notation) the Greek letter sigma (Σ) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it

a sum obtained by using the maximum value of $f (x)$ on each subinterval

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Calculus Volume 1 · 5.1 Approximating Areas

Approximating Areas: practice problems

32 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 1
State whether the given sums are equal or unequal.
$\sum_{i = 1}^{10} i$ and $\sum_{k = 1}^{10} k$
$\sum_{i = 1}^{10} i$ and
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Exercises 3
In the following exercises, use the rules for sums of powers of integers to compute the sums.
$\sum_{i = 5}^{10} i^{2}$
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Exercises 5
Suppose that $\sum_{i = 1}^{100} a_{i} = 15$ and $\sum_{i = 1}^{100} b_{i} = - 12.$ In the following exercises, compute the sums.
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Exercises 7
Suppose that $\sum_{i = 1}^{100} a_{i} = 15$ and $\sum_{i = 1}^{100} b_{i} = - 12.$ In the following exercises, compute the sums.
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Exercises 8
In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.
$\sum_{k = 1}^{20} 100 (k^{2} - 5 k + 1)$
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Exercises 9
In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.
$\sum_{j = 1}^{50} (j^{2} - 2 j)$
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Exercises 11
In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.
$\sum_{k = 1}^{25} [(2 k)^{2} - 100 k]$
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Exercises 13
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.
R₄ for $g (x) = cos (π x)$ on $[0, 1]$
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Exercises 15
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.
R₆ for $f (x) = \frac{1}{x ( x - 1 )}$ on
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Exercises 17
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
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Exercises 18
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 $x^{2} - 2 x + 1$ on
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Exercises 19
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 $x^{2} - 2 x + 1$ on
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Exercises 21
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.
Compute the left and right Riemann sums—L₆ and R₆, respectively—for $f (x) = (3 - ∣3 - x ∣)$ on Compute their average value and compare it with the area under the graph of .
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Exercises 23
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.
Compute the left and right Riemann sums—L₆ and R₆, respectively—for $f (x) = 9 - (x - 3)^{2}$ on and compare their values.
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Exercises 25
Express the following endpoint sums in sigma notation but do not evaluate them.
L₁₀ for $f (x) = 4 - x^{2}$ on $[- 2, 2]$
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Exercises 27
Express the following endpoint sums in sigma notation but do not evaluate them.
R₁₀₀ for $ln x$ on $[1, e]$
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Exercises 29
In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. If the two agree, say "neither."
[T] L₁₀₀ and R₁₀₀ for $y = x^{2}$ on the interval $[0, 1]$
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Exercises 31
In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. If the two agree, say "neither."
[T] L₁₀₀ and R₁₀₀ for $y = x^{3} + 2$ on the interval $[- 1, 1]$
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Exercises 33
In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. If the two agree, say "neither."
[T] L₁₀₀ and R₁₀₀ for $y = e^{2 x}$ on the interval $[- 1, 1]$
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Exercises 35
In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. If the two agree, say "neither."
Let $r_{j}$ denote the total rainfall in Portland on the jth day of the year in 2009. Interpret $\sum_{j = 1}^{31} r_{j} .$
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Exercises 37
In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. If the two agree, say "neither."
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?
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Exercises 39
In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. If the two agree, say "neither."
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.
Source: http://link.springer.com/article/10.1007%2Fs10712-011-9119-1
Starting Year 20-Year Change
1870 0.3
1890 1.5
1910 0.2
1930 2.8
1950 0.7
1970 1.1
1990 1.5
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Exercises 41
In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. If the two agree, say "neither."
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.
Source: http://www.census.gov/popest/data.
Year % Change/Year
2000 1.12
2001 0.99
2002 0.93
2003 0.86
2004 0.93
2005 0.93
2006 0.97
2007 0.96
2008 0.95
2009 0.88
(Hint: To obtain the population in July 2001, multiply the population in July 2000 by 1.0112 to get 284,573,831.)
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Exercises 43
In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L₈.
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Exercises 45
In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L₈.
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Exercises 47
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 $f (x) = \frac{1}{1 + x ^{2}}$ on
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Exercises 49
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] $y = cos (π x)$ on the interval
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Exercises 51
In the following exercises, use a calculator or a computer program to evaluate the endpoint sums R $_{N}$ and L $_{N}$ for $N = 1, 10, 100.$
[T] $y = x^{4} - 5 x^{2} + 4$ on the interval which has an exact area of
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Exercises 53
In the following exercises, use a calculator or a computer program to evaluate the endpoint sums R $_{N}$ and L $_{N}$ for $N = 1, 10, 100.$
Explain why, if $f (a) \geq 0$ and is increasing on that the left endpoint estimate is a lower bound for the area below the graph of on
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Exercises 55
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, $R_{N} - L_{N} = (b - a) \times \frac{f ( b ) - f ( a )}{N} .$
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Exercises 57
In the following exercises, use a calculator or a computer program to evaluate the endpoint sums R $_{N}$ and L $_{N}$ for $N = 1, 10, 100.$
For each of the three graphs:
Obtain a lower bound $L (A)$ for the area enclosed by the curve by adding the areas of the squares by the curve.
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Exercises 59
In the following exercises, use a calculator or a computer program to evaluate the endpoint sums R $_{N}$ and L $_{N}$ for $N = 1, 10, 100.$
A unit circle is made up of n wedges equivalent to the inner wedge in the figure. The base of the inner triangle is 1 unit and its height is
$sin (\frac{2 π}{n}) .$ The base of the outer triangle is
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Key terms · 8

left-endpoint approximation: an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle
lower sum: a sum obtained by using the minimum value of $f (x)$ on each subinterval
partition: a set of points that divides an interval into subintervals
regular partition: a partition in which the subintervals all have the same width
riemann sum: an estimate of the area under the curve of the form $A \approx \sum_{i = 1}^{n} f (x_i^*) \text{Δ} x$
right-endpoint approximation: the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle
sigma notation: (also, summation notation) the Greek letter sigma (Σ) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it
upper sum: a sum obtained by using the maximum value of $f (x)$ on each subinterval

Learn 5.1 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 5.1 in the textbook · This section on openstax.org

Approximating Areas: practice problems

32 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 1

State whether the given sums are equal or unequal.

$\sum_{i = 1}^{10} i$ and $\sum_{k = 1}^{10} k$
$\sum_{i = 1}^{10} i$ and

Exercises 3

In the following exercises, use the rules for sums of powers of integers to compute the sums.

$\sum_{i = 5}^{10} i^{2}$

Exercises 5

Suppose that $\sum_{i = 1}^{100} a_{i} = 15$ and $\sum_{i = 1}^{100} b_{i} = - 12.$ In the following exercises, compute the sums.

Exercises 7

Suppose that $\sum_{i = 1}^{100} a_{i} = 15$ and $\sum_{i = 1}^{100} b_{i} = - 12.$ In the following exercises, compute the sums.

Exercises 8

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.

$\sum_{k = 1}^{20} 100 (k^{2} - 5 k + 1)$

Exercises 9

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.

$\sum_{j = 1}^{50} (j^{2} - 2 j)$

Exercises 11

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.

$\sum_{k = 1}^{25} [(2 k)^{2} - 100 k]$

Exercises 13

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.

R₄ for $g (x) = cos (π x)$ on $[0, 1]$

Exercises 15

R₆ for $f (x) = \frac{1}{x ( x - 1 )}$ on

Exercises 17

L₄ for $\frac{1}{x ^{2} + 1}$ on

Exercises 18

R₈ for $x^{2} - 2 x + 1$ on

Exercises 19

L₈ for $x^{2} - 2 x + 1$ on

Exercises 21

Compute the left and right Riemann sums—L₆ and R₆, respectively—for $f (x) = (3 - ∣3 - x ∣)$ on Compute their average value and compare it with the area under the graph of .

Exercises 23

Compute the left and right Riemann sums—L₆ and R₆, respectively—for $f (x) = 9 - (x - 3)^{2}$ on and compare their values.

Exercises 25

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

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

Exercises 27

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

R₁₀₀ for $ln x$ on $[1, e]$

Exercises 29