The Definite Integral

Each particular Riemann sum depends on several things: the function $f$, the interval $[a,b]$, the partition $P$ of the interval, and the values chosen for $\mathrm{c}_{\mathrm{k}}$ in each subinterval. Fortunately, for most of the functions needed for applications, as the approximating rectangles get thinner (as the mesh of P approaches 0 and the number of subintervals gets bigger) the values of the Riemann sums approach the same value independently of the particular partition P and the points $\mathrm{c}_{\mathrm{k}}$ . For these functions, the LIMIT (as the mesh approaches 0) of the Riemann sums is the same number no matter how the $\mathrm{c}_{\mathrm{k}}$ are chosen.

This limit of the Riemann sums is the next big topic in calculus, the definite integral. Integrals arise throughout the rest of this book and in applications in almost every field that uses mathematics.

Definition: The Definite Integral

If $\lim _{m e s h \rightarrow 0}\left(\sum_{k=1}^{n} f\left(c_{k}\right) \cdot \Delta x_{k}\right)$ equals a finite number $I$

then $f$ is integrable on the interval $[a, b]$ .

The number $I$ is called the Definite Integral of $f$ on $[a,b]$ and is written $\int_{\mathrm{a}}^{\mathbf{b}} \mathrm{f}(x) \mathrm{d} x$.

The symbol $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x) \mathrm{d} x$ is read as "the integral from $a$ to $b$ of eff of $x$ dee or "the integral from $a$ to $b$ of $\mathrm{f}(x)$ with respect to $x$". The name of each piece of the symbol is shown in Fig. 1.

Example 1: Describe the area between the graph of $f(x)=1 / x$, the x–axis, and the vertical lines at $x=1$ and $x=5$ as a limit of Riemann sums and as a definite integral.

Solution: $\text { Area }=\lim _{m e s h \rightarrow 0}\left(\sum_{k=1}^{n} \frac{1}{c_{k}} \Delta x_{k}\right)=\int_{1}^{5} \frac{1}{x} \mathrm{~d} x \approx 1.609$
(from Table 2 in Section 4.1).

Practice 1: Describe the area between the graph of $\mathrm{f}(x)=\sin (x)$, the x–axis, and the vertical lines at $x=0$ and $x=\pi$ as a limit of Riemann sums and as a definite integral.

Example 2: Using the idea of area, determine the values of 

$\text { (a) } \lim _{\operatorname{mesh} \rightarrow 0}\left(\sum_{k=1}^{n}\left(1+c_{k}\right) \Delta x_{k}\right)$ on the interval $[1,3]$

$\text { (b) } \int_{0}^{4}(5-x) \mathrm{d} x$

$\text { (c) } \int_{-1}^{1} \sqrt{1-x^{2}} \mathrm{~d} x$

Solution: 

(a) represents the area between the graph of $\mathrm{f}(x)=1+x$ the x–axis, and the vertical lines at 1 and 3 (Fig. 2), and this area equals 6 square units.

b. represents the area between $\mathrm{f}(x)=5-x$, the x–axis and the vertical lines at 0 and 4, so the integral equals 12 square units.

c. represents the area of 1/2 of the circle $x^{2}+y^{2}=1$ with radius 1 and center at (0,0), and the integral equals $(circle area)/2 = \left(\pi r^{2}\right) / 2=\pi / 2$.

Practice 2: Using the area idea, determine the values of

a. $\lim _{\text {mesh } \rightarrow 0}\left(\sum_{k=1}^{n}\left(2 c_{k}\right) \Delta x_{k}\right)$ on the interval $[1,3]$ and (b) $\int_{3}^{8} 4 \mathrm{~d} x$.

Example 3: Represent the limit of each Riemann sum as a definite integral.

a. $\lim _{m e s h \rightarrow 0}\left(\sum_{k=1}^{n}\left(3+c_{k}\right) \Delta x_{k}\right)$ on $[1,4]$. 

b. $\lim _{\text {mesh } \rightarrow 0}\left(\sum_{k=1}^{n} \sqrt{c_{k}} \cdot \Delta x_{k}\right)$ on $[0,9]$.

Solution: (a) $\int_{1}^{4}(3+x) \mathrm{d} x$                    (b) $\int_{0}^{9} \sqrt{x} d x$

Example 4: Represent each shaded area in Fig. 3 as a definite integral. (Do not evaluate the definite integral, just translate the picture into symbols.)

Solution: (a) $\int_{-2}^{2}\left(4-x^{2}\right) \mathrm{d} x$               (b) $\int_{\pi / 2}^{\pi} \sin (x) \mathrm{d} x$

The value of a definite integral $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x) \mathrm{d} x$ depends only on the function $f$ being integrated and on the interval $[a, b]$. The variable $x \text { in } \int_{a}^{b} \mathrm{f}(x) \mathrm{d} x$ is a "dummy variable" and replacing it with another variable does not change the value of the integral. The following integrals each represent the integral of the function f on the interval $[a,b]$, and they are all equal:

$\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x) \mathrm{d} x=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(t) \mathrm{d} t \quad=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(w) \mathrm{d} w=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(z) \mathrm{d} z$

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.3-Definite-Integral.pdf
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