Definite Integrals

Geometric and Intuitive Meaning

In the broader chapter on integrals, the idea of “accumulating change” and “area under a curve” has already been introduced. The definite integral is the precise way we express that accumulation between two specific $x$–values.

For a function $f$ and real numbers $a < b$, the definite integral
$$
\int_a^b f(x)\,dx
$$
represents the net signed area between the graph of $y = f(x)$ and the $x$–axis from $x = a$ to $x = b$.

• Where $f(x) > 0$, the area contributes positively.
• Where $f(x) < 0$, the area contributes negatively.

So, if $f$ switches sign, the integral adds positive and negative contributions together, giving a net value, not just a total area.

Two important interpretations:

1. Net signed area
   $$\int_a^b f(x)\,dx$$ is “area above the axis minus area below the axis.”
2. Accumulated change
   If $f(x)$ is a rate of change of some quantity $F(x)$, then $\int_a^b f(x)\,dx$ is the total change of that quantity from $x = a$ to $x = b$.

You will see specific applications (such as distance, area, volume) in other chapters; here we focus on the basic properties and calculations of definite integrals themselves.

Riemann Sum Definition (Conceptual)

The “Riemann sums” subsection will treat this more systematically, but here is the core idea that makes a definite integral precise.

To define
$$
\int_a^b f(x)\,dx,
$$
we:

1. Split the interval $[a,b]$ into $n$ subintervals.
2. On each subinterval, choose a sample point $x_i^*$.
3. Form the sum
   $$
   \sum_{i=1}^n f(x_i^*)\,\Delta x_i,
   $$
   where $\Delta x_i$ is the width of the $i$‑th subinterval.
4. Let the maximum width of the subintervals go to $0$ (equivalently, let $n \to \infty$ in a controlled way).

When this limit exists and has the same value no matter how we choose the partition and sample points (as long as the partition gets finer), we define that limit to be the definite integral:
$$
\int_a^b f(x)\,dx = \lim_{\text{mesh} \to 0} \sum_{i=1}^n f(x_i^*)\,\Delta x_i.
$$

This connects the geometric picture (sum of many thin rectangles) with a rigorous limit.

Notation and Basic Conventions

A typical definite integral is written as
$$
\int_a^b f(x)\,dx.
$$

Each part has a role:

• $\int$ is the integral sign.
• $a$ and $b$ are the limits (or bounds) of integration.
• $f(x)$ is the integrand.
• $dx$ indicates the variable of integration.

Key conventions:

• The variable of integration is a dummy variable:
  $$
  \int_a^b f(x)\,dx = \int_a^b f(t)\,dt
  $$
  as long as the integrand is the same function, just with a different symbol.
• If $a = b$, then the integral is $0$:
  $$
  \int_a^a f(x)\,dx = 0.
  $$

Fundamental Algebraic Properties

Definite integrals have algebraic properties that allow you to break them apart, combine them, and transform them. These parallel many properties of sums and differences.

Assume $f$ and $g$ are integrable on $[a,b]$ and $k$ is a constant.

Linearity

1. Integral of a sum:
   $$
   \int_a^b (f(x) + g(x))\,dx = \int_a^b f(x)\,dx + \int_a^b g(x)\,dx.
   $$
2. Integral of a difference:
   $$
   \int_a^b (f(x) - g(x))\,dx = \int_a^b f(x)\,dx - \int_a^b g(x)\,dx.
   $$
3. Constant multiple:
   $$
   \int_a^b k\,f(x)\,dx = k \int_a^b f(x)\,dx.
   $$

These allow you to handle complicated integrands by splitting them into simpler ones.

Ordering and Inequalities

If $f(x) \le g(x)$ for all $x$ in $[a,b]$, then
$$
\int_a^b f(x)\,dx \le \int_a^b g(x)\,dx.
$$

If $f(x) \ge 0$ on $[a,b]$, then
$$
\int_a^b f(x)\,dx \ge 0.
$$

These reflect the geometric idea that “more area above the axis” gives a larger integral.

Additivity over Intervals

If $a < c < b$, then
$$
\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx.
$$

This captures the idea that the total accumulation from $a$ to $b$ can be split at any intermediate point $c$.

It is common to use this to evaluate integrals piecewise, especially when $f$ has different expressions on different subintervals.

Reversing the Limits

Swapping the limits reverses the sign:
$$
\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx.
$$

This is consistent with orientation: integrating “from left to right” is considered positive; going from right to left reverses the sign.

Definite Integrals and Antiderivatives

The Fundamental Theorem of Calculus (covered in detail in its own subsection) connects definite integrals with antiderivatives.

For now, we use this connection as a tool for computing definite integrals.

Suppose $F$ is an antiderivative of $f$ on $[a,b]$, meaning
$$
F'(x) = f(x)
$$
for all $x$ in that interval. Then
$$
\int_a^b f(x)\,dx = F(b) - F(a).
$$

This is often written using the evaluation bar:
$$
\int_a^b f(x)\,dx = \Bigl[\,F(x)\,\Bigr]_a^b = F(b) - F(a).
$$

The general strategy for computing a definite integral (when possible) is:

1. Find an antiderivative $F$ of $f$ (this uses techniques from indefinite integrals).
2. Evaluate $F(b) - F(a)$.

In practice, this allows one to avoid explicit Riemann sum limits in routine calculations.

Net Area vs. Geometric Area

The definite integral gives net signed area. Sometimes you need the actual geometric area, which is always nonnegative.

If $f(x)$ stays nonnegative on $[a,b]$, then the integral and the geometric area coincide:
$$
\text{Area} = \int_a^b f(x)\,dx.
$$

If $f(x)$ is sometimes negative, then:

• The definite integral:
  $$
  \int_a^b f(x)\,dx
  $$
  is area above the $x$–axis minus area below.
• The geometric area between the curve and the axis is:
  $$
  \int_a^b |f(x)|\,dx.
  $$

To compute this geometrically, one often:

1. Finds where $f(x) = 0$ (points where the curve crosses the axis).
2. Splits the interval accordingly.
3. Integrates $f(x)$ on parts where it is positive and $-f(x)$ where it is negative, then adds.

The detailed applications of this idea (e.g., plane areas) are handled in the “Applications of Integrals” chapter.

Symmetry and Definite Integrals

Symmetry can simplify definite integrals, especially over intervals symmetric about $0$ (like $[-a, a]$).

A function $f$ is:

• Even if $f(-x) = f(x)$ for all $x$ in its domain.
• Odd if $f(-x) = -f(x)$ for all $x$.

Assume $f$ is integrable on $[-a,a]$.

Even Functions

If $f$ is even, then
$$
\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx.
$$

Geometrically, the graph is symmetric with respect to the $y$–axis, so the area from $-a$ to $0$ equals the area from $0$ to $a$.

Odd Functions

If $f$ is odd, then
$$
\int_{-a}^a f(x)\,dx = 0.
$$

Geometrically, the area above the axis on the right is exactly canceled by the area below the axis on the left, due to origin symmetry.

These symmetry rules are useful shortcuts when they apply.

Change of Variable in Definite Integrals (Conceptual)

The “antiderivatives” chapter covers substitution for indefinite integrals. For definite integrals, a change of variable modifies both the integrand and the limits.

Suppose you make a substitution
$$
u = g(x)
$$
with $g$ differentiable and $g$ strictly increasing or decreasing on $[a,b]$. Then
$$
\int_a^b f(g(x))\,g'(x)\,dx
=
\int_{g(a)}^{g(b)} f(u)\,du.
$$

Key points specific to definite integrals:

• You do not need to go back to $x$ after integrating; instead, you change the limits from $x$–values to $u$–values.
• The new lower limit is $u = g(a)$, the new upper limit is $u = g(b)$.
• If $g$ is decreasing, the order of limits may flip; one can handle this either by reversing the limits and adding a minus sign, or just noting $g(a)$ and $g(b)$ may not be in increasing order.

Full methods and examples of substitution itself belong to the antiderivatives section; here the important idea is how the bounds transform.

Piecewise and Discontinuous Integrands

Definite integrals can be taken of functions that are not given by a single simple formula.

Piecewise Functions

If $f$ is defined by different formulas on subintervals, the integral is computed piece by piece using additivity:

If
$$
f(x) =
\begin{cases}
f_1(x) & a \le x < c,\\[4pt]
f_2(x) & c \le x \le b,
\end{cases}
$$
then
$$
\int_a^b f(x)\,dx
=
\int_a^c f_1(x)\,dx + \int_c^b f_2(x)\,dx.
$$

This is an immediate use of the additivity property and is very common in applications.

Discontinuities

A full discussion of continuity belongs to the “Continuity” subsection, but for definite integrals, the following general idea is important:

• Many functions with simple discontinuities (for example, a jump at a single point) are still integrable.
• Points where a function has a finite jump do not affect the value of the integral, because they influence only a single point and a single point has “zero width.”

However, if $f$ becomes unbounded or has infinitely many bad discontinuities on an interval, then the definite integral may fail to exist in the usual sense; such cases are treated under improper integrals, which are not the focus here.

Definite Integrals and Averages

Definite integrals can express an “average value” of a function over an interval.

If $f$ is integrable on $[a,b]$, its average value on that interval is defined by
$$
f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x)\,dx.
$$

This number has the property that the rectangle of height $f_{\text{avg}}$ and width $b-a$ has the same area as the signed area under the curve between $a$ and $b$:
$$
\int_a^b f(x)\,dx = f_{\text{avg}}(b-a).
$$

This concept is central in applications like average velocity, mean temperature, and many situations in statistics and physics, but the definition itself is a straightforward use of definite integrals.

Summary of Key Points

• A definite integral $\int_a^b f(x)\,dx$ gives the net signed area under $f(x)$ from $x = a$ to $x = b$, and more generally, the total accumulated change of a quantity whose rate is $f$.
• It is defined as the limit of Riemann sums and depends on the function and the interval, but not on the specific partition as long as it becomes arbitrarily fine.
• Algebraic properties (linearity, additivity over intervals, order reversal) make integrals manageable and parallel properties of sums.
• Antiderivatives allow you to compute definite integrals efficiently via $F(b) - F(a)$.
• Symmetry and change of variable provide powerful shortcuts for specific types of integrals.
• Piecewise functions are integrated by splitting the interval and combining results.
• The average value of a function over an interval is given by $\frac{1}{b-a}\int_a^b f(x)\,dx$.

Other chapters (Riemann sums, the Fundamental Theorem of Calculus, applications such as area and volume) build on these core ideas to extend what definite integrals can do and how they are computed in practice.