The definite integral and its properties

Definite integrals as sums of signed areas, integrals of odd and even functions over symmetric intervals, dissecting the interval, reversing the limits, sums and multiples of functions, and inequalities.

Signed areas

When the curve is below the $x$-axis, the integral comes out negative. In general, $\displaystyle\int_a^b f(x)\,dx$ is the area above the axis minus the area below it.

Example 1●

Evaluate $\displaystyle\int_0^3 (x - 1)\,dx$ and explain the answer with areas.

$\Big[\tfrac{x^2}{2} - x\Big]_0^3 = \tfrac92 - 3 = \tfrac32$.

There is a triangle of area $\tfrac12$ below the axis (from $0$ to $1$) and one of area $2$ above (from $1$ to $3$). And $2 - \tfrac12 = \tfrac32$.

Odd and even functions

An odd function cancels

Orange areas count as positive and blue as negative. For an odd function they are mirror images, so they cancel exactly.

Try this

Does the total area also come to $0$?

Example 2●●

Evaluate $\displaystyle\int_{-1}^{1} (x^4 + x^2)\,dx$.

The integrand is even, so it is $2\displaystyle\int_0^1 (x^4 + x^2)\,dx = 2\left(\tfrac15 + \tfrac13\right) = \tfrac{16}{15}$.

Rules for definite integrals

Dissection	$\displaystyle\int_a^b f\,dx + \int_b^c f\,dx = \int_a^c f\,dx$
Zero width	$\displaystyle\int_a^a f\,dx = 0$
Reversing	$\displaystyle\int_b^a f\,dx = -\int_a^b f\,dx$
Sums and multiples	$\displaystyle\int_a^b \big(kf + g\big)\,dx = k\int_a^b f\,dx + \int_a^b g\,dx$
Inequalities	If $f(x) \le g(x)$ on $[a, b]$, then $\displaystyle\int_a^b f\,dx \le \int_a^b g\,dx$

Example 3●●

Given $\displaystyle\int_0^3 f(x)\,dx = 5$ and $\displaystyle\int_0^1 f(x)\,dx = 2$, find $\displaystyle\int_1^3 f(x)\,dx$, $\;\displaystyle\int_3^0 f(x)\,dx$ and $\displaystyle\int_0^3 \big(2f(x) + 1\big)\,dx$.

Dissection: $\displaystyle\int_1^3 f\,dx = 5 - 2 = 3$.

Reversing: $\displaystyle\int_3^0 f\,dx = -5$.

$2 \times 5 + \displaystyle\int_0^3 1\,dx = 10 + 3 = 13$.

Flashcards

Odd $f$, from $-a$ to $a$

$\displaystyle\int_{-a}^{a} f(x)\,dx = 0$

Even $f$, from $-a$ to $a$

$2\displaystyle\int_0^a f(x)\,dx$

$displaystyleint_b^a f,dx$

$-\displaystyle\int_a^b f\,dx$

Practice

Evaluate $\displaystyle\int_0^3 (x - 1)\,dx$.
Evaluate $\displaystyle\int_{-2}^{2} (x^3 - 4x)\,dx$.
Evaluate $\displaystyle\int_{-1}^{1} (x^4 + x^2)\,dx$.
$\displaystyle\int_0^3 f = 5$ and $\displaystyle\int_0^1 f = 2$. Find $\displaystyle\int_1^3 f$.
For the same $f$, find $\displaystyle\int_3^0 f(x)\,dx$.
For the same $f$, find $\displaystyle\int_0^3 \big(2f(x) + 1\big)\,dx$.
Evaluate $\displaystyle\int_{-2}^{2} x^2\,dx$.
$\displaystyle\int_4^4 (x^3 + 7)\,dx = $
A negative definite integral means: