Definite Integrals

In the previous chapter, the integral sign $\int$ meant "find a function whose derivative is the integrand." That is the indefinite integral, and it produces a family of functions. In this chapter, the same symbol acquires a second, seemingly unrelated meaning: a limit of sums of rectangle areas, which produces a single number. That number is the definite integral.

The two ideas look so different that using one symbol for both seems like an abuse of notation. The Fundamental Theorem of Calculus shows that it is not. Differentiation and integration are inverse operations, and the theorem lets us evaluate a definite integral, defined as a limit of increasingly complicated sums, by the simple recipe

\int_a^b f(x)\,dx = F(b) - F(a),\qquad\text{where } F' = f.

That single formula converts a hard limit problem into an antidifferentiation problem, and it is the reason integral calculus is usable at all.

Sections in This Chapter

The order matters here more than in most chapters. Section 7.1 defines the object, Section 7.2 gives the algebraic rules it obeys, and Section 7.3 supplies the computational tool that the remaining four sections use constantly. Sections 7.4 through 7.7 are four independent consequences of that tool.

The Central Idea in One Picture

Suppose you want the area between a curve $y=f(x)$ and the $x$-axis from $x=a$ to $x=b$. You cannot use a formula from elementary geometry, because the top boundary is curved. So you approximate:

1. Cut $[a,b]$ into $n$ thin vertical strips.
2. Replace each strip by a rectangle whose height is the value of $f$ at some sample point in that strip.
3. Add the rectangle areas. This sum is called a Riemann sum.
4. Let the strips get thinner and thinner. If the sums converge to a single number regardless of how the strips and sample points were chosen, that number is the definite integral.

Every idea in this chapter is a refinement of those four steps. Notice, in particular, step 4: the definition is a limit, and limits are where the analysis lives.

One warning that pays off later: the definite integral measures net area, area above the $x$-axis counted positively and area below it counted negatively. It is not the same as the total geometric area unless $f$ never dips below the axis.

Where Definite Integrals Are Used

The area interpretation is the easiest one to picture, but it is far from the most important. Whenever a quantity accumulates, a definite integral computes the total.

• Distance from velocity. If $v(t)$ is the velocity of a moving object, then ${\int }_{t_1}^{t_2}v(t)dt$ is the net displacement over that time interval, and ${\int }_{t_1}^{t_2}|v(t)|dt$ is the total distance travelled. This is the cleanest example of the difference between net and total.
• Net change of any quantity. If Q'(t) is the rate at which a quantity changes, then \int_a^b Q'(t)\,dt = Q(b) - Q(a). Water flowing into a tank, a population growing, a bank balance accruing interest: all are the same statement of the Fundamental Theorem.
• Work in physics. A variable force $F(x)$ acting along a line does work ${\int }_a^{b}F(x)dx$. Compressing a spring, pumping water out of a tank, and lifting a hanging cable are standard applications.
• Average value. The mean temperature over a day, the root-mean-square voltage of an alternating current, and the average concentration of a drug in the bloodstream are all defined by $\bar{f}=\frac{1}{b-a}{\int }_a^{b}f$, the subject of Section 7.7.
• Probability. For a continuous random variable with density $f$, the probability that the variable falls in $[a,b]$ is exactly ${\int }_a^{b}f(x)dx$, and the expected value is $\int xf(x)dx$.
• Mass and centre of mass. A rod of variable density $\rho (x)$ has mass ${\int }_a^b\rho (x)dx$, and its balance point is found from a closely related integral.
• Functions defined by integrals. Some of the most useful functions in mathematics, including $\ln x={\int }_1^x\frac{dt}{t}$, the error function, and the Fresnel integrals, have no elementary antiderivative and are defined as definite integrals with a variable upper limit. Section 7.4 shows how to differentiate them.

How to Read This Chapter

Section 7.1 is the conceptual foundation, and it is also the most computationally tedious. The limit-of-Riemann-sums calculations there are done once, to establish what the integral means, and then essentially never again, because Section 7.3 replaces them with a two-line evaluation.

It is worth resisting the temptation to skip ahead to that shortcut. Every subtlety that appears later, why continuity is required, why a vertical asymptote breaks the evaluation rule, why the substitution rule changes the limits of integration, traces back to the definition as a limit. Read Section 7.1 for meaning, and Section 7.3 for method.