Active Calculus

That these limits always exist (and share the same value) when $f$ is continuous

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It turns out that a function need not be continuous in order to have a definite integral. For our purposes, we assume that the functions we consider are continuous on the interval(s) of interest. It is straightforward to see that any function that is piecewise continuous on an interval of interest will also have a well-defined definite integral.

allows us to make the following definition.

We call the symbol $\int$ the integral sign, the values $a$ and $b$ the limits of integration, and the function $f$ the integrand. The process of determining the real number ${\int }_a^{b}f(x)dx$ is called evaluating the definite integral. While there are several different interpretations of the definite integral, for now the most important is that ${\int }_a^{b}f(x)dx$ measures the net signed area bounded by $y=f(x)$ and the $x$-axis on the interval $[a,b]\text{.}$

If the velocity function is nonnegative on $[a,b]\text{,}$ then ${\int }_a^{b}v(t)dt$ tells us the distance the object traveled. If the velocity is sometimes negative on $[a,b]\text{,}$ we can use definite integrals to find the areas bounded by the function on each interval where $v$ does not change sign, and the sum of these areas will tell us the distance the object traveled.

To compute the value of a definite integral from the definition, we have to take the limit of a sum. While this is possible to do in select circumstances, it is also tedious and time-consuming, and does not offer much additional insight into the meaning or interpretation of the definite integral. Instead, in Section 4.4, we will learn the Fundamental Theorem of Calculus, which provides a shortcut for evaluating a large class of definite integrals. This will enable us to determine the exact net signed area bounded by a continuous function and the $x$-axis in many circumstances.

For now, our goal is to understand the meaning and properties of the definite integral, rather than to compute its value. To do this, we will rely on the net signed area interpretation of the definite integral. So we will use as examples curves that produce regions whose areas we can compute exactly through area formulas. We can thus compute the exact value of the corresponding integral.

Regarding the definite integral of a function $f$ over an interval $[a,b]$ as the net signed area bounded by $f$ and the $x$-axis, we discover several standard properties of the definite integral. It is helpful to remember that the definite integral is defined in terms of Riemann sums, which consist of the areas of rectangles.

For any real number $a$ and the definite integral ${\int }_a^{a}f(x)dx$ it is evident that no area is enclosed, because the interval begins and ends with the same point. Hence,

While this rule is easy to see if $a<b<c\text{,}$ it in fact holds in general for any values of $a\text{,}$ $b\text{,}$ and $c\text{.}$ Another property of the definite integral states that if we reverse the order of the limits of integration, we change the sign of the integral’s value.

This result makes sense because if we integrate from $a$ to $b\text{,}$ then in the defining Riemann sum we set $\mathrm{\Delta }x=\frac{b-a}{n}\text{,}$ while if we integrate from $b$ to $a\text{,}$ we have $\mathrm{\Delta }x=\frac{a-b}{n}=-\frac{b-a}{n}\text{,}$ and this is the only change in the sum used to define the integral.

These rules are useful because they allow to deal individually with the simplest parts of certain functions by taking advantage of addition and multiplying by a constant. In other words, the process of taking the derivative respects addition and multiplying by constants in the simplest possible way.

It turns out that similar rules hold for the definite integral. First, let’s consider the functions pictured in Figure 4.3.7.

Because multiplying the function by 2 doubles its height at every $x$-value, we see that the height of each rectangle in a left Riemann sum is doubled, $f(x_i)$ for the original function, versus $2f(x_i)$ in the doubled function. For the areas $A$ and $B\text{,}$ it follows $B=2A\text{.}$ As this is true regardless of the value of $n$ or the type of sum we use, we see that in the limit, the area of the red region bounded by $y=2f(x)$ will be twice the area of the blue region bounded by $y=f(x)\text{.}$ As there is nothing special about the value $2$ compared to an arbitrary constant $k\text{,}$ the following general principle holds.

We see a similar situation with the sum of two functions $f$ and $g\text{.}$

If we take the sum of two functions $f$ and $g$ at every point in the interval, the height of the function $f+g$ is given by $(f+g)(x_i)=f(x_i)+g(x_i)\text{.}$ Hence, for the pictured rectangles with areas $A\text{,}$ $B\text{,}$ and $C\text{,}$ it follows that $C=A+B\text{.}$ Because this will occur for every such rectangle, in the limit the area of the gray region will be the sum of the areas of the blue and red regions. In terms of definite integrals, we have the following general rule.

Solving Equation (4.3.2) for $f_{\mathrm{AVG}[a,b]}\text{,}$ we have the following general principle.

Equation (4.3.2) tells us another way to interpret the definite integral: the definite integral of a function $f$ from $a$ to $b$ is the length of the interval $(b-a)$ times the average value of the function on the interval. In addition, when the function $f$ is nonnegative on $[a,b]\text{,}$ Equation (4.3.2) has a natural visual interpretation.

Consider Figure 4.3.9, where we see at left the shaded region whose area is ${\int }_a^{b}f(x)dx\text{,}$ at center the shaded rectangle whose dimensions are $(b-a)$ by $f_{\mathrm{AVG}[a,b]}\text{,}$ and at right these two figures superimposed. Note that in dark green we show the horizontal line $y=f_{\mathrm{AVG}[a,b]}\text{.}$ Thus, the area of the green rectangle is given by $(b-a)\cdot f_{\mathrm{AVG}[a,b]}\text{,}$ which is precisely the value of ${\int }_a^{b}f(x)dx\text{.}$ The area of the blue region in the left figure is the same as the area of the green rectangle in the center figure. We can also observe that the areas $A_1$ and $A_2$ in the rightmost figure appear to be equal. Thus, knowing the average value of a function enables us to construct a rectangle whose area is the same as the value of the definite integral of the function on the interval. This applet

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provides an opportunity to explore how the average value of the function changes as the interval changes, through an image similar to that found in Figure 4.3.9.