The area between a graph and the x-axis, and between two graphs

Exponential and logarithmic functions, grade 12 (4 units) Lesson 18 of 20Learning

The area between a graph and the x-axis, and between two graphs

An area above the

\displaystyle x

-axis

Where the graph of $\displaystyle f$ is above the $\displaystyle x$ -axis for $\displaystyle a\le x\le b$ , the area between the graph and the axis is the definite integral $\displaystyle \int_a^b f(x)\,dx$ . The example below calculates the area bounded by the graph of $\displaystyle {f(x)}={e^x}$ , the $\displaystyle x$ -axis and the lines $\displaystyle {x}={0}$ and $\displaystyle {x}={1}$ :

\displaystyle f(x)=e^{x}

S=\int_0^1e^x\,dx=\left[e^x\right]_0^1=e-1\approx1.72

An area below the

\displaystyle x

-axis

Where the graph is below the $\displaystyle x$ -axis, the integral is negative, and the area is its opposite – or the integral of $\displaystyle -f(x)$ . The graph of $\displaystyle {f(x)}={e^x-3}$ crosses the $\displaystyle x$ -axis at $\displaystyle {x}={\ln3}$ , and to the left of that point it is below the axis. The example below calculates the area bounded by the graph, the $\displaystyle x$ -axis and the $\displaystyle y$ -axis:

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

S=\int_0^{\ln3}\left(3-e^x\right)dx=\left[3x-e^x\right]_0^{\ln3}=(3\ln3-3)-(0-1)=3\ln3-2\approx1.30

A function that changes sign

Where the function changes sign in the interval, we split the interval at its zero and calculate each part on its own. The function $\displaystyle {f(x)}={\frac{2}{x}-1}$ is zero at $\displaystyle {x}={2}$ : between 1 and 2 it is positive, between 2 and 4 negative.

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

The part above the axis: $\displaystyle {\int_1^2\left(\frac{2}{x}-1\right)dx}={\left[2\ln x-x\right]_1^2}={2\ln2-1}$ .
The part below the axis: $\displaystyle {\int_2^4\left(1-\frac{2}{x}\right)dx}={\left[x-2\ln x\right]_2^4}={2-2\ln2}$ .
The whole area: $\displaystyle {(2\ln2-1)+(2-2\ln2)}={1}$ .
The integral from 1 to 4 without splitting gives $\displaystyle 4\ln2-3\approx-0.23$ : the parts cancel each other out, and that is not the area.

The area between two graphs

The area between two graphs, in an interval where $\displaystyle f(x)\ge g(x)$ :

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

The example below calculates the area bounded by the graphs of $\displaystyle {f(x)}={e^{3x}-12e^{2x}+36e^x}$ and of $\displaystyle {g(x)}={e^{3x}}$ and the $\displaystyle y$ -axis:

\displaystyle f(x)=e^{3x}-12e^{2x}+36e^{x}

\displaystyle g(x)=e^{3x}

The meeting point: $\displaystyle {f(x)}={g(x)}$ gives $\displaystyle {36e^x-12e^{2x}}={0}$ , that is $\displaystyle {12e^x(3-e^x)}={0}$ . $\displaystyle e^x>0$ , so $\displaystyle {e^x}={3}$ and $\displaystyle {x}={\ln3}$ .
For $\displaystyle 0\le x\le\ln3$ : $\displaystyle {f(x)-g(x)}={36e^x-12e^{2x}\ge0}$ .
The area:

S=\int_0^{\ln3}\left(36e^x-12e^{2x}\right)dx=\left[36e^x-6e^{2x}\right]_0^{\ln3}=(108-54)-(36-6)=24

A parameter from a given area

When the area is given, a limit of the integral can be the unknown. The example below finds $\displaystyle a>1$ such that the area bounded by the graph of $\displaystyle {f(x)}={\frac{1}{x}}$ , the $\displaystyle x$ -axis and the lines $\displaystyle {x}={1}$ and $\displaystyle {x}={a}$ is 2:

$\displaystyle {\int_1^a\frac{1}{x}\,dx}={\left[\ln x\right]_1^a}={\ln a-\ln1}={\ln a}$ .
$\displaystyle {\ln a}={2}$ , so $\displaystyle {a}={e^2\approx7.39}$ .

Exercise 1

Calculate the area bounded by the graph of $\displaystyle {f(x)}={e^{2x}}$ , the $\displaystyle x$ -axis and the lines $\displaystyle {x}={0}$ and $\displaystyle {x}={1}$ .

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

/ – 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

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

Mark the points where the graph meets the axes.

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

Calculate the area bounded by the graph of the function, the $\displaystyle x$ -axis and the $\displaystyle y$ -axis.

/ – 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 functions $\displaystyle {f(x)}={\frac{3}{x}}$ and $\displaystyle {g(x)}={4-x}$ are given, for $\displaystyle x>0$ .

Find the $\displaystyle x$ -coordinates of the points where the two graphs meet.

Calculate the area enclosed between the two graphs.

\displaystyle f(x)=\frac{3}{x}

\displaystyle g(x)=4-x

/ – 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 4

The function $\displaystyle {f(x)}={1-\frac{1}{x}}$ is given.

Find the zero of the function.

Calculate the area bounded by the graph of the function, the $\displaystyle x$ -axis and the lines $\displaystyle {x}={\frac{1}{2}}$ and $\displaystyle {x}={2}$ .

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

/ – 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 5

$\displaystyle k$ is a positive parameter.
The area bounded by the graph of $\displaystyle {f(x)}={e^x}$ , the axes and the line $\displaystyle {x}={k}$ is 3.
Find $\displaystyle k$ : write an equation in $\displaystyle k$ and solve it.

Formulas: exponential and logarithmic functions, grade 12 (4 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals and the area

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int e^{mx+b}\,dx=\frac{e^{mx+b}}{m}+C$ $\displaystyle m\neq0$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle \int\frac{1}{mx+b}\,dx=\frac{\ln|mx+b|}{m}+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$