3.1.1 Introduction
The purpose of this chapter is to compute the area of the region enclosed by the graph of a function \(y=f(x)\), \(x \in [a,b]\), and the \(x\)-axis, under the assumption that \(f(x)\geq 0\) for all \(x \in [a,b]\). We will do this for polynomial functions. The idea is to approximate such area by means of the area of rectangular regions, as in Figure 3.1.
![(a) Graph of a function above the x-axis over an interval [a,b] with the region enclosed by the graph and the axis shaded](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-f01a.png)
Figure 3.1(a) Area under the Graph of f
Figure 3.1(b) Rectangular Approximation
We shall restrict our attention to a special type of approximations. Namely, right-endpoint rectangular approximations. The goal of this section is to define such approximations and introduced the notation that will be used throughout the chapter.
3.1.2 Right-Endpoint Rectangular Approximations
A right-endpoint rectangular approximation satisfies the following two conditions:
- All rectangles have the same width.
- The upper right-vertex of each rectangle lies on the graph of \(f\).
We will denote the value of the sum of the areas of the rectangles by \(R(f,n)\), where \(n\) is the number of rectangles in the approximation. We will also use \(R(f,n)\) to denote the graphical representation of the rectangular approximation.
Example 1. The pictures in Figure 3.2 show right-endpoint approximations for a given function \(f\) defined on the interval \([-2,4]\). Pay attention to the location of the upper right vertices of the rectangles.
![(a) Graph of a positive function on [-2,4], with two rectangles of widths of 3 units along the x axis, upper right vertices at (1,f(1)) and (4, f(4))](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-f02a.png)
Figure 3.2(a) R(f,2) = 27
Figure 3.2(b) R(f,3) = 26
Figure 3.2(c) R(f,6) = 25
On Your Own 1. Given the function \(f\) as in Figure 3.3, draw \(R(f,n)\) for \(n=2\), \(3\), and \(6\), and find its value in each case.
![the region enclosed by a new positive function on [-2,4] for you to draw rectangular approximations with 2 rectangles](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-f03a.png)
Figure 3.3(a) R(f,2) =
![the region enclosed by a new positive function on [-2,4] for you to draw rectangular approximations with 3 rectangles](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-f03b.png)
Figure 3.3(b) R(f,3) =
![the region enclosed by a new positive function on [-2,4] for you to draw rectangular approximations with 6 rectangles](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-f03c.png)
Figure 3.3(c) R(f,6) =
3.1.3 Notation.
The following notation related with right-endpoint rectangular approximations will be used through out this chapter and in subsequent chapters. Thus, make sure to become fully familiar with it.
Let \(f\) be a continuous function defined on \( x\in [a,b]\) such that
\[f(x)\geq 0 \;\; \forall x\in [a,b].\]
- The interval \([a,b]\) is divided into \(n\) subintervals of equal length.
This is accomplished by constructing a partition of the interval \([a,b]\) into \(n\) subintervals of equal length. The length of the intervals is denoted by \(\Delta x\), and is given by \(\displaystyle \Delta x=\frac{b-a}{n}\). - The endpoints of a partition of an interval \([a,b]\) are denoted as follows:\(x_0=a=\) left endpoint of the interval \([a,b]\);\(x_n=b=\) right endpoint of the interval \([a,b]\).\begin{align*}
&x_0=a\\
&x_1=x_0+\Delta x,\\
&x_2=x_1+\Delta x =x_0+2\Delta x,\\
&x_3=x_2+\Delta x=x_0+3\Delta x,\\
&\hphantom{==} \vdots\\
& x_i=x_{i-1}+ \Delta x=x_0+i \Delta x,\\
&\hphantom{==} \vdots\\
& x_n =x_{n-1}+\Delta x = x_0+n \Delta x=b
\end{align*}
For example, in Figure 3.2(a),
\[[a,b]=[-2,4],\, n=2, \Delta x = \frac{4-(-2)}{2}=3, \, x_0=-2, x_1=-2+ 3=1,\, x_2=-2+2(3)=4.\]Note that there are \(n+1\) points in the partition of the interval, because we start at \(x_0 = a\) and end at \(x_n = b\), thus, the \(n+1\) points are: \(x_0, \, x_1,\, x_2,\, \dots , , x_{n-1}, \, x_n\). - Notation for the \(n\) intervals:
\[I_1=\left[x_0,x_1\right], I_2=\left[x_1,x_2\right], \dots, I_i=\left[x_{i-1},x_i\right],
\dots, I_{n-1}=\left[x_{n-2},x_{n-1}\right], I_n=\left[x_{n-1},x_n\right].\]
In particular, the \(i\)-th interval is given by \(I_i=\left[x_{i-1},x_i\right]\).The picture in Figure 3.4 shows the \(n\) intervals of the partition together with the \(n+1\) endpoints. The distance between two consecutive endpoints is the length of the intervals, which is \(\Delta x\).![Real number line with interval [a,b] partitioned into n subintervals. Endpoints of the intervals highlighted in red and denoted x1, x2, ..., xn](https://sites.psu.edu/math140jaj/files/2019/08/ch03-01-f04-300x25.png)
Figure 3.4: Partition of the interval [a; b] into n subintervals of equal length.
- The rectangles are denoted by \(R_1\), \(R_2, \,\dots,\, R_n\), with \(R_i\) the \(i\)-th rectangle with base the interval \([x_{i-1},x_i]\).
- \(h_i\) denotes the height of \(R_i\). This height is given by
\[h_i=f(x_i), \;\;\; i =1,2,\dots, n.\]
Namely, the height of each rectangle is the \(y\)-coordinate of the upper right-vertex of the rectangle. - \(A_i= A(R_i)\) denotes the area of the \(i\)-th rectangle. This is given by
\[A_i= f(x_i)\Delta x, \;\;\; i =1,2,\dots, n.\]
![Real number line with interval [a,b] partitioned into n subintervals. Endpoints of the intervals highlighted in red and denoted x1, x2, ..., xn](https://sites.psu.edu/math140jaj/files/2019/08/ch03-01-f04.png){kind=small}
In the following example, no grid lines are given, thus, it will be necessary to compute the heights of the rectangular regions explicitly using the \(y\)-coordinates of the points on the graph of \(f\).
Example 2. Let \(f(x)=-x^2+2x+4\), \(x\in [-1,2]\). Find \(R(f,3)\) and draw the rectangular approximation. (See Figure 3.5.)
![(a) Graph of the parabola f(x) = -x^2+2x+4 over the interval [-1,2], with region enclosed by the graph and the x axis shaded](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-f05a.png)
Figure 3.5(a) Example 2 \((fx)=-x^2+2x\)
Figure 3.5(b) Example 2 R(f,3) = 13
Solution.
(1) Partition the interval into three subintervals of equal length, \(\displaystyle \Delta x=\frac{2-(-1)}{3}=1\).
Points of the partition: \(x_0=-1,\;\; x_1=-1+1=0,\;\; x_2=-1+2(1)=1,\;\; x_3=-1+3(1)=2\).
(2) Height and area of each rectangular region.
\[
\renewcommand{\arraystretch}{1.5}
\begin{array}{c|l|l}
\text{Rectangle} & \hspace{.6in}\text{Height} & \hspace{.5in}\text{Area}\\ \hline
R_1 &
h_1=f\left(x_1\right)=f\left(0\right)=4&
A_1=f\left(x_1\right)\Delta x=4\cdot 1\\
R_2 &
h_2=f\left(x_2\right)=f(1)=5&
A_2=f\left(x_2\right)\Delta x=5\cdot 1\\
R_3 &
h_3=f\left(x_3\right)=f\left(2\right)=4&
A_3=f\left(x_3\right)\Delta x=4\cdot 1
\end{array}
\]
(3) Sum of the areas of the rectangular regions: \(R(f,3)=4\cdot 1+5\cdot 1+4\cdot 1=13\).
On Your Own 2. Let \(f(x)=4-x^2\), \(x\in [-2,2]\). Find \(R(f,4)\) and draw the rectangular approximation.
Example 3. Let \(f(x)=\cos x + 1\), \(\displaystyle x\in \left[-\frac{\pi}{2},\pi \right]\). Find \(R(f,6)\).
Solution. Figure 3.6{a} shows \(R(f,6)\).
(1) Partition the interval into six subintervals of equal length, \(\displaystyle \Delta x=\frac{\pi -(-\pi /2)}{6}=\frac{\pi }{4}\).
Endpoints of the partition:
\[
\renewcommand{\arraystretch}{2.5}
\begin{array}{lll}
\displaystyle x_0= a=-\frac{\pi }{2} & &\\
\displaystyle x_1=-\frac{\pi }{2}+\frac{\pi }{4}=-\frac{\pi }{4},
&\displaystyle x_2=-\frac{\pi }{2}+2\left(\frac{\pi }{4}\right)=0,
&\displaystyle x_3=-\frac{\pi }{2}+3\left(\frac{\pi}{4}\right)=\frac{\pi }{4},\\
\displaystyle x_4=-\frac{\pi }{2}+4\left(\frac{\pi }{4}\right)=\frac{\pi }{2},
&\displaystyle x_5=-\frac{\pi }{2}+5\left(\frac{\pi }{4}\right)=\frac{3\pi }{4},
&\displaystyle x_6=b=-\frac{\pi}{2}+6\left(\frac{\pi }{4}\right)=\pi.
\end{array}
\]
(2) Height and area of each rectangular region.
\[
\renewcommand{\arraystretch}{2.5}
\begin{array}{c|l|l}
\text{Rectangle} & \hspace{.8in}\text{Height} & \hspace{.75in}\text{Area}\\ \hline
R_1 &\displaystyle
h_1=f\left(x_1\right)=f\left(-\frac{\pi }{4}\right)=1+\frac{\sqrt{2}}{2}&\displaystyle
A_1=f\left(x_1\right)\Delta x=\left(1+\frac{\sqrt{2}}{2}\right)\cdot \frac{\pi}{4}\\
R_2 &\displaystyle
h_2=f\left(x_2\right)=f(0)=2&\displaystyle
A_2=f\left(x_2\right)\Delta x=\left(2\right)\cdot \frac{\pi}{4}\\
R_3 &\displaystyle
h_3=f\left(x_3\right)=f\left(\frac{\pi }{4}\right)=1+\frac{\sqrt{2}}{2}&\displaystyle
A_3=f\left(x_3\right)\Delta x=\left(1+\frac{\sqrt{2}}{2}\right)\cdot \frac{\pi}{4}\\
R_4 &\displaystyle
h_4=f\left(x_4\right)=f\left(\frac{\pi }{2}\right)=1&\displaystyle
A_4=f\left(x_4\right)\Delta x=\left(1\right)\cdot \frac{\pi}{4}\\
R_5 &\displaystyle
h_5=f\left(x_5\right)=f\left(\frac{3\pi }{4}\right)=1-\frac{\sqrt{2}}{2}&\displaystyle
A_5=f\left(x_5\right)\Delta x=\left(1-\frac{\sqrt{2}}{2}\right)\cdot \frac{\pi}{4}\\
R_6 &\displaystyle
h_6=f\left(x_6\right)=f(\pi)=0&\displaystyle
A_6=f\left(x_6\right)\Delta x=\left(0\right)\cdot \frac{\pi}{4}
\end{array}
\]
(3) Sum of the areas of the rectangular regions.
\begin{align*}
R(f,6)&=\left(1+\frac{\sqrt{2}}{2}\right)\left(\frac{\pi }{4}\right)+(2)\left(\frac{\pi }{4}\right)+\left(1+\frac{\sqrt{2}}{2}\right)\left(\frac{\pi
}{4}\right)+(1)\left(\frac{\pi }{4}\right)+ \\
&\hspace{.5in} +\left(1-\frac{\sqrt{2}}{2}\right)\left(\frac{\pi }{4}\right)+(0)\left(\frac{\pi }{4}\right)=\left(\frac{12+\sqrt{2}}{2}\right)\frac{\pi }{4}.
\end{align*}
On Your Own 3. Let \(f(x)=\sin x +1\), \(\displaystyle x\in \left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). Find \(R(f,3)\) and draw the rectangular approximation.
![(a) Graph of f(x) = cos x +1, x in interval [-pi/2, pi] with right-endpoint rectangular approximation with 6 rectangles](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-f06a.png)
Figure 3.6(a) Example 3
![(b) Graph of f(x) = arcsin x +pi/2, x in interval [-1,1] with right-endpoint rectangular approximation with 4 rectangles](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-f06b.png)
Figure 3.6(b) Example 4
Example 4. Let \(\displaystyle f(x)=\arcsin x + \frac{\pi}{2}\), \( x\in \left[-1,1 \right]\). Find \(R(f,4)\).
Solution. Figure 3.5(b) shows \(R(f,4)\).
(1) Partition the interval into 4 subintervals of equal length,
\[\Delta x=\frac{1-(-1))}{4}=\frac{1 }{2}.\]
The endpoints of the partition are given by \( a=x_0=-1\), \(\displaystyle x_1=-\frac{1}{2}\), \( x_2=0\), \(\displaystyle x_3=\frac{1}{2}\), and \(b = x_4 = 1.\)
(2) Height and area of each rectangular region.
\[\renewcommand{\arraystretch}{2.5}
\begin{array}{c|l|l}
\text{Rectangle} & \hspace{.8in}\text{Height} & \hspace{.75in}\text{Area}\\ \hline
R_1 &\displaystyle
h_1=\arcsin\left(-\frac{1}{2}\right)+\frac{\pi}{2}=-\frac{\pi}{6}+\frac{\pi}{2}=\frac{\pi}{3}&\displaystyle
A_1=f\left(x_1\right)\Delta x=\frac{\pi}{3}\cdot \frac{1}{2}=\frac{\pi}{6}\\
R_2 &\displaystyle
h_2=\arcsin\left(0\right)+\frac{\pi}{2}=0+\frac{\pi}{2}=\frac{\pi}{2}&\displaystyle
A_2=f\left(x_2\right)\Delta x=\frac{\pi}{2}\cdot \frac{1}{2}=\frac{\pi}{4}\\
R_3 &\displaystyle
h_3=\arcsin\left(\frac{1}{2}\right)+\frac{\pi}{2}=\frac{\pi}{6}+\frac{\pi}{2}=\frac{2\pi}{3}&\displaystyle
A_3=f\left(x_3\right)\Delta x=\frac{2\pi}{3}\cdot \frac{1}{2}=\frac{\pi}{3}\\
R_4 &\displaystyle
h_4=\arcsin\left(1\right)+\frac{\pi}{2}=\frac{\pi}{1}+\frac{\pi}{2}=\pi&\displaystyle
A_4=f\left(x_4\right)\Delta x=\pi\cdot \frac{1}{2}=\frac{\pi}{2}
\end{array}
\]
(3) Sum of the areas of the rectangular regions.
\[R(f,4)=\frac{\pi}{6} + \frac{\pi}{4} + \frac{\pi}{3} +\frac{\pi}{2} = \frac{2\pi+3\pi + 4\pi +6\pi}{12} =\frac{15\pi}{12} =\frac{5\pi}{4}.\]
3.1.4 Exercises
A
-
- For each of the graphs in the figure below, draw the the indicated approximation and find its value.
![(a) Graph of a positive function over interval [-2,4] with shaded region enclosed by the graph and the x axis. You need to draw R(f,3)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq01a.png)
Exercise 1(a) R(f,3) =
![(b) Graph of a positive function over interval [-4,4] with shaded region enclosed by the graph and the x axis. You need to draw R(f,4)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq01b.png)
Exercise 1(b) R(f,4) =
![(c) Graph of a positive function over interval [-3,3] with shaded region enclosed by the graph and the x axis. You need to draw R(f,3)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq01c.png)
Exercise 1(c) M(f,3) =
![(d) Graph of a positive function over interval [-2,2] with shaded region enclosed by the graph and the x axis. You need to draw R(f,2)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq01d.png)
Exercise 1(d) R(f,2) =
In problems 2–11, for each functions, sketch its graph and \(R(f,n)\), and find its exact value.
- \(f(x)=4x-x^2\), \(0\leq x\leq 4\); \(R(f,4)\).
- \(f(x)=\arccos x\), \(-1\leq x\leq 1\); \(R(f,4)\).
- \(f(x)=\tan x + 2\), \(\displaystyle -\frac{\pi }{3}\leq x\leq \frac{\pi }{3}\); \(R(f,4)\).
- \(f(x)=\sec x \), \(\displaystyle -\frac{\pi }{3}\leq x\leq \frac{\pi }{3}\); \(R(f,4)\).
- \(f(x)=e^x \), \([-2,2]\); \(R(f,4)\). Hint: for your final answer factor.
- \(f(x)=\ln x \), \([e,6e]\); \(R(f,5)\). Hint: prime factorization.
- \(f(x)=e^x \), \(\displaystyle \left [ \frac{1}{2},\frac{3}{2}\right ]\); \(R(f,4)\).
- \(f(x)=\ln x \), \([\sqrt{e},8\sqrt{e}]\); \(R(f,7)\).
- \(\displaystyle f(x)=\arcsin x +\frac{\pi}{2}\), \(\displaystyle \left[-\frac{\sqrt{3}}{2},
\frac{\sqrt{3}}{2}\right]\); \(R(f,2)\). - \(\displaystyle f(x)=\arctan x +\frac{\pi}{2}\), \(\left[-2,1\right]\); \(R(f,3)\).
B Midpoint and Left-Endpoint Approximations
-
-
- Write a brief paragraph explaining how to draw and compute \(R(f,n)\) for a function defined on an interval \([a,b]\), and \(n \in \mathbb{N}\).
- Let \(M(f,n)\) denote the midpoint rectangular approximation. Write a brief paragraph explaining how to draw and compute \(M(f,n)\) for a function defined on \([a,b]\), and \(n \in \mathbb{N}\).
- For each of the graphs in the figure below, draw the the indicated approximation and find its value.
![(a) Graph of a positive function over interval [-2,4] with shaded region enclosed by the graph and the x axis. You need to draw M(f,3)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq15a-300x232.png)
Exercise 15(a) M(f,3) =
![(b) Graph of a positive function over interval [-4,4] with shaded region enclosed by the graph and the x axis. You need to draw M(f,4)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq15b-300x232.png)
Exercise 15(b) M(f,4) =
![(c) Graph of a positive function over interval [-3,3] with shaded region enclosed by the graph and the x axis. You need to draw M(f,3)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq15c-300x232.png)
Exercise 15(c) M(f,3) =
![(d) Graph of a positive function over interval [-2,2] with shaded region enclosed by the graph and the x axis. You need to draw M(f,2)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq15d-232x300.png)
Exercise 15(d) M(f,2) =
- Let \(L(f,n)\) denote the left-endpoint rectangular approximation. Write a brief paragraph explaining how to draw and compute \(L(f,n)\) for a function defined on \([a,b]\), and \(n \in \mathbb{N}\).
- For each of the graphs in the figure below, draw the the indicated approximation and find its value.
![(a) Graph of a positive function over interval [-2,4] with shaded region enclosed by the graph and the x axis. You need to draw L(f,3)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq17a-300x232.png)
Exercise 17(a) L(f,3) =
![(b) Graph of a positive function over interval [-4,4] with shaded region enclosed by the graph and the x axis. You need to draw L(f,4)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq17b-300x232.png)
Exercise 17(b) L(f,4) =
![(c) Graph of a positive function over interval [-3,3] with shaded region enclosed by the graph and the x axis. You need to draw L(f,3)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq17c-300x232.png)
Exercise 17(c) L(f,3) =
![(d) Graph of a positive function over interval [-2,2] with shaded region enclosed by the graph and the x axis. You need to draw L(f,2)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq17d-232x300.png)
Exercise 17(d) L(f,2) =
-
![(a) Graph of a positive function over interval [-2,4] with shaded region enclosed by the graph and the x axis. You need to draw M(f,3)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq15a.png)
![(b) Graph of a positive function over interval [-4,4] with shaded region enclosed by the graph and the x axis. You need to draw M(f,4)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq15b.png)
![(c) Graph of a positive function over interval [-3,3] with shaded region enclosed by the graph and the x axis. You need to draw M(f,3)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq15c.png)
![(d) Graph of a positive function over interval [-2,2] with shaded region enclosed by the graph and the x axis. You need to draw M(f,2)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq15d.png)
![(a) Graph of a positive function over interval [-2,4] with shaded region enclosed by the graph and the x axis. You need to draw L(f,3)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq17a.png)
![(b) Graph of a positive function over interval [-4,4] with shaded region enclosed by the graph and the x axis. You need to draw L(f,4)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq17b.png)
![(c) Graph of a positive function over interval [-3,3] with shaded region enclosed by the graph and the x axis. You need to draw L(f,3)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq17c.png)
![(d) Graph of a positive function over interval [-2,2] with shaded region enclosed by the graph and the x axis. You need to draw L(f,2)](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-fq17d.png)
In problems 17 and 18, (a) sketch the graph of \(f\); (b) draw and compute (i) \(M(f,n)\); (ii) \(L(f,n)\).
- \(f(x)=2\sin x + 3\), \(x\in [-\pi,\pi]\); \(n=4\).
- \(f(x)=-x^2+4 x+5\), \(x\in [-1,5]\); \(n=3\).
- Given \(f(x)=\cos x +2\), \(\displaystyle \left[-\frac{3\pi}{4},-\frac{\pi}{4}\right]\), (a) sketch its graph;(b) draw and compute the exact value of (i) \(R(f,3)\); (ii) \(L(f,3)\). Hint: use the subtraction formula for cosine.
Answers to On Your Own
Exercise 1(a) R(f,2) = 6
Exercise 1(b) R(f,3) = 6
Exercise 1(c) R(f,6) = 6
![# 2 Right-endpoint rectangular approximation, with 3 rectangles, to the region enclosed by the graph of f(x) = 4 -x^2, x in [-2,2]](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-faoyo02a.png)
Exercise 2 R(f,4) = 10
![# 3 Right-endpoint rectangular approximation, with 3 rectangles, to the region enclosed by the graph of f(x) = sin x +1, x in [-pi/2,pi/2]](https://sites.psu.edu/math140jaj/files/2019/12/ch03-01-faoyo03a.png)
Exercise 3 R(f,3) = \(\frac{4\pi}3\)