The definite integral and its properties

Integral calculus, grade 11 (5 units) Lesson 7 of 20Learning

The definite integral and its properties

Calculating a definite integral

Calculate a definite integral in three steps: an antiderivative, putting in the limits, the difference. The example below calculates $\displaystyle \int_{-1}^{2}\left(x^2-2x\right)dx$ :

An antiderivative: $\displaystyle \frac{x^3}{3}-x^2$ .
$\displaystyle {\left[\frac{x^3}{3}-x^2\right]_{-1}^{2}}={\left(\frac{8}{3}-4\right)-\left(-\frac{1}{3}-1\right)}={-\frac{4}{3}+\frac{4}{3}}={0}$ .
The integral is 0, but there is area: the area above the axis, between $\displaystyle -1$ and 0, equals the area below the axis, between 0 and 2.

\displaystyle f(x)=x^{2}-2x

The properties of the definite integral

The properties of the definite integral follow from $\displaystyle {\int_a^b f(x)\,dx}={F(b)-F(a)}$ , and from the areas too:

\displaystyle \int_{a}^{b}k\cdot f(x)\,dx

\displaystyle =

\displaystyle k\int_{a}^{b}f(x)\,dx

\displaystyle \int_{a}^{b}\left(f(x)+g(x)\right)dx

\displaystyle =

\displaystyle \int_{a}^{b}f(x)\,dx+\int_{a}^{b}g(x)\,dx

\displaystyle \int_{a}^{b}f(x)\,dx+\int_{b}^{c}f(x)\,dx

\displaystyle =

\displaystyle \int_{a}^{c}f(x)\,dx

\displaystyle \int_{b}^{a}f(x)\,dx

\displaystyle =

\displaystyle -\int_{a}^{b}f(x)\,dx

From $\displaystyle a$ to $\displaystyle a$ : $\displaystyle {\int_a^a f(x)\,dx}={F(a)-F(a)}={0}$ .
Where $\displaystyle f(x)\geq0$ throughout the interval ( $\displaystyle a<b$ ), the integral is not negative: it is the area.
Where $\displaystyle f(x)\leq0$ throughout, the integral is minus the area.

Integrals from a given area

Sometimes there is no formula, only an area, and integrals can still be calculated with the properties. In the example below the graph of $\displaystyle f$ is above the $\displaystyle x$ -axis for $\displaystyle 2\le x\le6$ , and the area under it is 40, that is $\displaystyle {\int_2^6 f(x)\,dx}={40}$ :

\displaystyle f(x)=\frac{15}{16}x(7-x)

$\displaystyle {\int_2^6\left(f(x)+5\right)dx}={40+\int_2^6 5\,dx}={40+5\cdot4}={60}$ : the graph went up by 5, and a rectangle was added.
$\displaystyle {\int_2^6\left(f(x)-3\right)dx}={40-3\cdot4}={28}$ .
$\displaystyle {\int_2^6 -f(x)\,dx}={-40}$ : the graph was reflected in the $\displaystyle x$ -axis.
$\displaystyle {\int_0^4 f(x+2)\,dx}={40}$ : the graph of $\displaystyle f(x+2)$ is the graph of $\displaystyle f$ shifted 2 to the left, and the interval $\displaystyle 0\le x\le4$ moved with it.
$\displaystyle \int_2^6 f(x+2)\,dx$ cannot be calculated: it is the area under $\displaystyle f$ between 4 and 8, and more information is needed.

Exercise 1

Calculate the integrals.

a

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

Watch the order of the limits.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Given: $\displaystyle {\int_0^3 f(x)\,dx}={7}$ and $\displaystyle {\int_3^5 f(x)\,dx}={-2}$ .
Calculate:

a

$\displaystyle \int_0^5 f(x)\,dx$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

$\displaystyle \int_5^3 f(x)\,dx$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

$\displaystyle \int_0^3\left(2f(x)+x\right)dx$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

$\displaystyle k>0$ is a number for which $\displaystyle {\int_0^k\left(2x-1\right)dx}={6}$ .

a

Calculate the integral as an expression in $\displaystyle k$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Find $\displaystyle k$ .

Formulas: integral calculus, grade 11 (5 units)

Basic integrals

The indefinite integral – all the antiderivatives
$\displaystyle \int f(x)\,dx=F(x)+C$ $\displaystyle F^{\prime}(x)=f(x)$
A power
$\displaystyle \int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
A constant
$\displaystyle \int k\,dx=kx+C$
$\displaystyle \int\frac{1}{x^{2}}\,dx=-\frac{1}{x}+C$
One over a root
$\displaystyle \int\frac{1}{\sqrt{x}}\,dx=2\sqrt{x}+C$
A root
$\displaystyle \int\sqrt{x}\,dx=\frac{2}{3}x\sqrt{x}+C$

The rules of integration

A constant factor
$\displaystyle \int k\cdot f(x)\,dx$ $\displaystyle =$ $\displaystyle k\int f(x)\,dx$
A sum and a difference
$\displaystyle \int\left(f(x)\pm g(x)\right)dx$ $\displaystyle =$ $\displaystyle \int f(x)\,dx\pm\int g(x)\,dx$
A linear inner function
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
A composite function times the inner derivative
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
A power of a function times its derivative
$\displaystyle \int\left(u(x)\right)^{n}\cdot u^{\prime}(x)\,dx=\frac{\left(u(x)\right)^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$

Trigonometric functions (in radians)

$\displaystyle \int\cos x\,dx=\sin x+C$
$\displaystyle \int\sin x\,dx=-\cos x+C$
$\displaystyle \int\frac{1}{\cos^{2}x}\,dx=\tan x+C$
From $\displaystyle {\cos2x}={1-2\sin^2x}$
$\displaystyle \sin^{2}x$ $\displaystyle =$ $\displaystyle \frac{1-\cos2x}{2}$
From $\displaystyle {\cos2x}={2\cos^2x-1}$
$\displaystyle \cos^{2}x$ $\displaystyle =$ $\displaystyle \frac{1+\cos2x}{2}$
From $\displaystyle {\sin2x}={2\sin x\cos x}$
$\displaystyle \sin x\cos x$ $\displaystyle =$ $\displaystyle \frac{1}{2}\sin2x$
From $\displaystyle {\sin^2x+\cos^2x}={1}$
$\displaystyle \tan^{2}x$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^{2}x}-1$

The accumulation function and the definite integral

The accumulation function of $\displaystyle f$ from $\displaystyle a$
$\displaystyle F_{a}(x)=\int_{a}^{x}f(t)\,dt$
The fundamental theorem
$\displaystyle F_{a}^{\prime}(x)=f(x)$
The definite integral, $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The same limit
$\displaystyle \int_{a}^{a}f(x)\,dx=0$
Swapping the limits
$\displaystyle \int_{b}^{a}f(x)\,dx$ $\displaystyle =$ $\displaystyle -\int_{a}^{b}f(x)\,dx$
Joining intervals
$\displaystyle \int_{a}^{b}f(x)\,dx+\int_{b}^{c}f(x)\,dx$ $\displaystyle =$ $\displaystyle \int_{a}^{c}f(x)\,dx$

Areas and accumulation

An area above the $\displaystyle x$ -axis
$\displaystyle S=\int_{a}^{b}f(x)\,dx$ $\displaystyle f(x)\geq0$
An area below the $\displaystyle x$ -axis
$\displaystyle S=-\int_{a}^{b}f(x)\,dx$ $\displaystyle f(x)\leq0$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The displacement from the velocity
$\displaystyle s(b)-s(a)=\int_{a}^{b}v(t)\,dt$