Areas and accumulation

Exponential and logarithmic functions, grade 12 (5 units) Lesson 21 of 23Learning

Areas and accumulation

An area above the

\displaystyle x

-axis

When 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 $\displaystyle {f(x)}={2^x}$ , the $\displaystyle x$ -axis and the lines $\displaystyle {x}={0}$ and $\displaystyle {x}={2}$ :

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

S=\int_0^2 2^x\,dx=\left[\frac{2^x}{\ln2}\right]_0^2=\frac{4-1}{\ln2}=\frac{3}{\ln2}\approx4.33

A function that changes sign

Where the graph is below the $\displaystyle x$ -axis the integral is negative, and the area is the integral of $\displaystyle -f(x)$ . When the function changes sign in the interval, split the interval at its zero and calculate each part separately. The example below calculates the area bounded by the graph of $\displaystyle {f(x)}={e^x-2}$ , the $\displaystyle x$ -axis and the lines $\displaystyle {x}={0}$ and $\displaystyle {x}={\ln4}$ :

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

The zero: $\displaystyle {e^x-2}={0}$ , that is $\displaystyle {e^x}={2}$ and $\displaystyle {x}={\ln2}$ .
Between 0 and $\displaystyle \ln2$ the graph is below the axis: $\displaystyle {\int_0^{\ln2}\left(2-e^x\right)dx}={\left[2x-e^x\right]_0^{\ln2}}={(2\ln2-2)-(0-1)}={2\ln2-1}$ .
Between $\displaystyle \ln2$ and $\displaystyle \ln4$ the graph is above the axis: $\displaystyle {\int_{\ln2}^{\ln4}\left(e^x-2\right)dx}={\left[e^x-2x\right]_{\ln2}^{\ln4}}={(4-2\ln4)-(2-2\ln2)}={2-2\ln2}$ .
The whole area: $\displaystyle {(2\ln2-1)+(2-2\ln2)}={1}$ .

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 $\displaystyle {f(x)}={e^x}$ and of $\displaystyle {g(x)}={e^{-x}}$ and the line $\displaystyle {x}={1}$ :

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

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

\displaystyle x=1

The meeting point: $\displaystyle {e^x}={e^{-x}}$ gives $\displaystyle {x}={-x}$ , that is $\displaystyle {x}={0}$ .
For $\displaystyle 0\le x\le1$ : $\displaystyle e^x\ge e^{-x}$ .
The area:

S=\int_0^1\left(e^x-e^{-x}\right)dx=\left[e^x+e^{-x}\right]_0^1=\left(e+\frac{1}{e}\right)-2\approx1.09

A composite area

When the upper edge of the area passes from one graph to another, split the area where they meet. The example below calculates the area bounded by the graphs of $\displaystyle {f(x)}={e^x}$ and of $\displaystyle {g(x)}={e^{2-x}}$ , the $\displaystyle x$ -axis and the lines $\displaystyle {x}={0}$ and $\displaystyle {x}={2}$ :

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

\displaystyle g(x)=e^{2-x}

The meeting point: $\displaystyle {e^x}={e^{2-x}}$ gives $\displaystyle {x}={2-x}$ , that is $\displaystyle {x}={1}$ .
Between 0 and 1 the area is under the graph of $\displaystyle f$ : $\displaystyle {\int_0^1e^x\,dx}={e-1}$ .
Between 1 and 2 the area is under the graph of $\displaystyle g$ : $\displaystyle {\int_1^2e^{2-x}\,dx}={\left[-e^{2-x}\right]_1^2}={-1+e}={e-1}$ .
The whole area: $\displaystyle {2(e-1)}={2e-2\approx3.44}$ .

An amount accumulated from a rate

When a function describes a rate – an amount per unit of time – its integral between two times is the amount accumulated between them. The example below calculates how much gold was produced in the world:

The time $\displaystyle t$ is measured in years, and $\displaystyle {t}={0}$ is the end of 2000.
The function $\displaystyle {p(t)}={2500e^{0.006t}}$ is a model of the rate of the world’s gold production, in tonnes a year.
At the end of 2000 the rate was $\displaystyle {p(0)}={2500}$ tonnes a year.
In 2001 the world produced:

\int_0^1 2500e^{0.006t}\,dt=\left[\frac{2500}{0.006}e^{0.006t}\right]_0^1=\frac{2500}{0.006}\left(e^{0.006}-1\right)\approx2507.5

The rate rose during the year, so the amount is a little more than 2500 tonnes.
In the ten years 2001–2010 the world produced $\displaystyle {\int_0^{10}2500e^{0.006t}\,dt}={\frac{2500}{0.006}\left(e^{0.06}-1\right)\approx25{,}765}$ tonnes.

The accumulation function of

\displaystyle \frac{1}{t}

In grade 11 you learnt that the area accumulation function $\displaystyle {F(x)}={\int_a^x f(t)\,dt}$ is an antiderivative of $\displaystyle f$ : $\displaystyle {F'(x)}={f(x)}$ . The accumulation function of $\displaystyle \frac{1}{t}$ from 1 is the natural logarithm:

F(x)=\int_1^x\frac{1}{t}\,dt=\left[\ln t\right]_1^x=\ln x-\ln1=\ln x

On the graph: the area under $\displaystyle \frac{1}{x}$ between 1 and $\displaystyle x$ is the height of $\displaystyle \ln x$ at that $\displaystyle x$ .

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

\displaystyle g(x)=\ln x

$\displaystyle {\ln e}={1}$ : the area under $\displaystyle \frac{1}{x}$ between 1 and $\displaystyle e$ is 1.
Where $\displaystyle 0<x<1$ the upper limit is less than 1, the integral is negative, and $\displaystyle \ln x<0$ too.
$\displaystyle {F'(x)}={\frac{1}{x}>0}$ , so $\displaystyle \ln x$ increases throughout its domain.

Exercise 1

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

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

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

Mark the zero of the function.

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

Calculate the area bounded by the graph, 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

Calculate the area bounded by the graph, the $\displaystyle x$ -axis and the line $\displaystyle {x}={2}$ .

/ – 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

Calculate the whole shaded area, between $\displaystyle {x}={0}$ and $\displaystyle {x}={2}$ .

/ – 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 $\displaystyle {f(x)}={\frac{4}{x}}$ and $\displaystyle {g(x)}={5-x}$ 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{4}{x}

\displaystyle g(x)=5-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 functions $\displaystyle {f(x)}={e^{2x}}$ and $\displaystyle {g(x)}={3e^x-2}$ are given.

Find the $\displaystyle x$ -coordinates of the points where the two graphs meet.
Substitute $\displaystyle {t}={e^x}$ .

Between the meeting points the graph of $\displaystyle g$ is above the graph of $\displaystyle f$ .
Calculate the area enclosed between the two graphs.

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

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

/ – 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

Water flows into a tank at a rate of $\displaystyle {r(t)}={20e^{-0.1t}}$ litres a minute.
$\displaystyle t$ is the time in minutes since the tap was opened, and the tank is empty when it is opened.

How many litres flow into the tank in the first 10 minutes?
Write an exact result, and on another line round it to one decimal place.

/ – 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

After $\displaystyle t$ minutes, $\displaystyle 200-200e^{-0.1t}$ litres have flowed into the tank.
After how many minutes will the tank hold 100 litres?

Exercise 6

The accumulation function $\displaystyle {F(x)}={\int_1^x\frac{2}{t}\,dt}$ is given, for $\displaystyle x>0$ .

Calculate $\displaystyle F(x)$ .

/ – 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

$\displaystyle {F(x)}={2\ln x}$ .
Find the $\displaystyle x$ at which $\displaystyle {F(x)}={4}$ .

Formulas: exponential and logarithmic functions, grade 12 (5 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
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
Change of base
$\displaystyle \log_{c}x$ $\displaystyle =$ $\displaystyle \frac{\log_{a}x}{\log_{a}c}$
$\displaystyle \log_{a}b$ $\displaystyle =$ $\displaystyle \frac{1}{\log_{b}a}$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

The exponential and the logarithmic function

Every exponential as a power of $\displaystyle e$
$\displaystyle a^{x}$ $\displaystyle =$ $\displaystyle e^{x\ln a}$
$\displaystyle a^x$ and $\displaystyle \log_a x$ – inverse functions
$\displaystyle y=a^{x}\iff x=\log_{a}y$ $\displaystyle y>0$
The number $\displaystyle e$
$\displaystyle e\approx2.718$

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 $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$
The doubling time $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle 2$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (a^{x})^{\prime}$ $\displaystyle =$ $\displaystyle a^{x}\cdot\ln a$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
$\displaystyle (\log_{a}x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x\cdot\ln a}$
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, the area and accumulation

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int a^{x}\,dx=\frac{a^{x}}{\ln a}+C$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
Where $\displaystyle f(x)\neq0$
$\displaystyle \int\frac{f^{\prime}(x)}{f(x)}\,dx=\ln|f(x)|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The accumulation function’s derivative
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$