Properties of the Definite Integral
Textbook Reference
| Primary source | OpenStax Calculus Volume 2, Section 1.2: “The Definite Integral” |
| Direct link | https://openstax.org/books/calculus-volume-2/pages/1-2-the-definite-integral |
| Textbook used in class | Stewart, Calculus, Section 4.2: “The Definite Integral” |
Opening Scenario
Computing $\displaystyle\int_0^5 (3x^2 - 2x + 7)\,dx$ from the limit definition would require three separate summation formulas and careful algebra. Instead, a small set of properties -- inherited directly from the Riemann sum definition -- lets you split, scale, and combine integrals. These properties are not shortcuts; they are exact equalities that follow from the definition.
Quick Reference
Let $f$ and $g$ be integrable on $[a, b]$ and let $c$ be a constant.
Convention: $\displaystyle\int_a^a f(x)\,dx = 0$. $\quad$ $\displaystyle\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx$.
Constant multiple: $\displaystyle\int_a^b c\,f(x)\,dx = c\int_a^b f(x)\,dx$.
Sum/difference: $\displaystyle\int_a^b [f(x) \pm g(x)]\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx$.
Additivity over intervals: $\displaystyle\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx$.
Comparison: If $f(x) \geq g(x)$ on $[a, b]$, then $\displaystyle\int_a^b f(x)\,dx \geq \int_a^b g(x)\,dx$.
Bound: If $m \leq f(x) \leq M$ on $[a, b]$, then $m(b-a) \leq \displaystyle\int_a^b f(x)\,dx \leq M(b-a)$.
Key Concepts
1. Linearity
The sum and constant-multiple rules together say that integration is linear: $$\int_a^b [c_1 f(x) + c_2 g(x)]\,dx = c_1 \int_a^b f(x)\,dx + c_2 \int_a^b g(x)\,dx.$$
This follows directly from the linearity of finite sums: $\sum (c_1 a_i + c_2 b_i) = c_1 \sum a_i + c_2 \sum b_i$. Since each Riemann sum has this property, so does the limit.
Practical use. Break a complicated integrand into simpler pieces, integrate each piece, and combine.
2. Interval Additivity
$\displaystyle\int_a^c f(x)\,dx = \int_a^b f(x)\,dx + \int_b^c f(x)\,dx$ for any $b$ between $a$ and $c$ (or even outside $[a, c]$ if the integrals exist).
Geometrically: the total signed area from $a$ to $c$ equals the signed area from $a$ to $b$ plus the signed area from $b$ to $c$. The point $b$ can be anywhere -- it just splits the region.
Practical use. If $f$ changes sign at $x = b$, compute $\int_a^b$ and $\int_b^c$ separately, then add. This is necessary to find total (unsigned) area.
3. Reversed Limits
$$\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx.$$
The definition assigns a sign based on orientation. Swapping the limits negates the integral. This convention ensures that the additivity property holds even when $b$ is not between $a$ and $c$.
4. Comparison and Bounding
If $f(x) \geq 0$ on $[a, b]$, then $\displaystyle\int_a^b f(x)\,dx \geq 0$. More generally, if $m \leq f(x) \leq M$ on $[a, b]$, then the integral is trapped between $m(b-a)$ and $M(b-a)$.
This bounding property is useful for estimating integrals when an exact antiderivative is difficult to find.
“You can factor a function of $x$ out of an integral.” The constant-multiple rule allows $\int_a^b c\,f(x)\,dx = c\int_a^b f(x)\,dx$ when $c$ does not depend on $x$. You cannot write $\int_a^b x\cdot f(x)\,dx = x\int_a^b f(x)\,dx$ because $x$ is the variable of integration, not a constant.
Worked Example
Given $\displaystyle\int_0^5 f(x)\,dx = 8$ and $\displaystyle\int_0^5 g(x)\,dx = 3$, evaluate:
(a) $\displaystyle\int_0^5 [2f(x) - 4g(x)]\,dx$
By linearity: $2(8) - 4(3) = 16 - 12 = 4$.
(b) $\displaystyle\int_5^0 f(x)\,dx$
By the reversed-limits rule: $-\displaystyle\int_0^5 f(x)\,dx = -8$.
Boxed answers: (a) $4$; (b) $-8$.
Given $\displaystyle\int_1^3 f(x)\,dx = 5$ and $\displaystyle\int_3^7 f(x)\,dx = -2$, find $\displaystyle\int_1^7 f(x)\,dx$.
By additivity: $5 + (-2) = 3$.
Boxed answer: $\displaystyle\int_1^7 f(x)\,dx = 3$.
Common Errors Summary
| Error | Example | Correction |
|---|---|---|
| Factoring a variable out of an integral | $\int_0^1 x e^x\,dx = x\int_0^1 e^x\,dx$ | $x$ is the integration variable; only constants can be factored out |
| Thinking $\int_a^b f \cdot g = \int_a^b f \cdot \int_a^b g$ | Splitting a product into a product of integrals | Integrals do not distribute over products (integration by parts handles products instead) |
| Forgetting the sign reversal when limits swap | Writing $\int_3^1 f\,dx = \int_1^3 f\,dx$ | $\int_3^1 f\,dx = -\int_1^3 f\,dx$ |
Leveled Practice
Level 1 -- Direct Application
Problem 1. If $\displaystyle\int_0^3 f(x)\,dx = 10$, find $\displaystyle\int_0^3 5f(x)\,dx$.
Show answer
$5 \times 10 = 50$.
Level 2 -- Combining Properties
Problem 2. Given $\displaystyle\int_0^2 f = 4$, $\displaystyle\int_2^5 f = -1$, and $\displaystyle\int_0^5 g = 6$, evaluate $\displaystyle\int_0^5 [f(x) + 2g(x)]\,dx$.
Show answer
$\int_0^5 f = 4 + (-1) = 3$ (additivity). $\int_0^5 2g = 2(6) = 12$.
$\displaystyle\int_0^5 [f + 2g]\,dx = 3 + 12 = 15$.
Level 3 -- Bounding
Problem 3. Use the bound property to estimate $\displaystyle\int_0^{\pi/2} \sin x\,dx$ without computing it.
Show answer
On $[0, \pi/2]$, $\sin x$ ranges from $0$ to $1$. So $m = 0$, $M = 1$, $b - a = \pi/2$.
$0 \leq \displaystyle\int_0^{\pi/2} \sin x\,dx \leq \frac{\pi}{2} \approx 1.57$.
(The exact value is $1$, which is indeed between 0 and 1.57.)
Mastery Checklist
Mental Model
The properties of the definite integral mirror the properties of ordinary sums because the integral is a limit of sums. Scaling every term of a sum by $c$ scales the total by $c$. Adding two sequences and summing is the same as summing each sequence and adding the totals. Splitting a sum at an intermediate index and adding both parts recovers the original total. Each property of the integral is just its summation counterpart, taken to the limit.
Connections
Looking back
- Linearity of sums: The sigma-notation rules $\sum(ca_i) = c\sum a_i$ and $\sum(a_i + b_i) = \sum a_i + \sum b_i$ pass through the limit to become integral rules.
Looking ahead
- FTC applications (Section 4.3): Linearity makes polynomial integrals easy to compute term by term.
- Substitution (Section 4.5): Interval splitting via additivity is used when the substitution changes the direction of the bounds.