The area between a graph and the x-axis

Integral calculus, grade 11 (5 units) Lesson 16 of 20Learning

The area between a graph and the x-axis

Above the axis and below it

An area is always positive, but the integral of a negative function is negative. So:

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

\displaystyle f(x)\geq0

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

\displaystyle f(x)\leq0

When the function changes sign in the interval, find its zeros, split the interval at them, and add the areas of the parts.
One integral over the whole interval gives the difference between the area above the axis and the area below it – not the area.

A function that changes sign

The example below calculates the area between the graph of $\displaystyle {f(x)}={x^2-1}$ and the $\displaystyle x$ -axis, for $\displaystyle 0\le x\le2$ :

\displaystyle f(x)=x^{2}-1

The zero in the interval: $\displaystyle {x}={1}$ . Between 0 and 1 the graph is below the axis, and between 1 and 2 above it.
$\displaystyle {\int_0^1\left(x^2-1\right)dx}={\left[\frac{x^3}{3}-x\right]_0^1}={-\frac{2}{3}}$ : the area is $\displaystyle \frac{2}{3}$ .
$\displaystyle {\int_1^2\left(x^2-1\right)dx}={\left(\frac{8}{3}-2\right)-\left(\frac{1}{3}-1\right)}={\frac{4}{3}}$ .
The whole area: $\displaystyle {\frac{2}{3}+\frac{4}{3}}={2}$ . (One integral from 0 to 2 would give $\displaystyle {-\frac{2}{3}+\frac{4}{3}}={\frac{2}{3}}$ .)

An enclosed area

“The area enclosed by the graph and the $\displaystyle x$ -axis” is the area between two neighbouring zeros – the limits are not given, so find them. In the example below $\displaystyle {f(x)}={3x-x^2}$ :

The zeros: $\displaystyle {3x-x^2}={0}$ , that is $\displaystyle {x}={0}$ or $\displaystyle {x}={3}$ .
Between 0 and 3 the graph is above the axis: $\displaystyle {S}={\int_0^3\left(3x-x^2\right)dx}={\left[\frac{3x^2}{2}-\frac{x^3}{3}\right]_0^3}={\frac{27}{2}-9}={\frac{9}{2}}$ .

\displaystyle f(x)=3x-x^{2}

Exercise 1

The function $\displaystyle {f(x)}={x^3-4x}$ is given.

a

Mark the points where the graph meets the $\displaystyle x$ -axis.

\displaystyle f(x)=x^{3}-4x

b

Calculate the area enclosed by the graph and the $\displaystyle x$ -axis for $\displaystyle 0\le x\le2$ .

/ – 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

What is the whole shaded area? What is $\displaystyle \int_{-2}^2f(x)\,dx$ ? Explain.

The shaded area

/ – 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

\displaystyle \int_{-2}^2f(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

Exercise 2

Calculate the area bounded by the graph of $\displaystyle {f(x)}={\cos x}$ , the $\displaystyle x$ -axis, the $\displaystyle y$ -axis and the line $\displaystyle {x}={\pi}$ .

\displaystyle f(x)=\cos x

/ – 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

The function $\displaystyle {f(x)}={\frac{1}{x^2}-1}$ is given, for $\displaystyle x>0$ .
Calculate the area bounded by the graph, the $\displaystyle x$ -axis and the lines $\displaystyle {x}={\frac{1}{2}}$ and $\displaystyle {x}={2}$ .

\displaystyle f(x)=\frac{1}{x^{2}}-1

/ – 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

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
$\displaystyle s(b)-s(a)=\int_{a}^{b}v(t)\,dt$