In the last lesson we encountered the notation for the area under a graph, and in our investigation on the fundamental theorem of calculus we have seen the connection between differential calculus, integral calculus and the area under a curve. Let's now look at how to use integral calculus to calculate the areas under a curve.
From our previous lesson we know that if we are given a continuous function $f$f on the interval $\left[a,b\right]$[a,b], where $f\left(x\right)\ge0$f(x)≥0 for all $x$x in the interval, then the value of the area under the graph of $y=f\left(x\right)$y=f(x) from $x=a$x=a to $x=b$x=b is called the definite integral and is written:
That is, for a function such as $f\left(x\right)$f(x) shown in the diagram above, where $f\left(x\right)\ge0$f(x)≥0 on the interval $a\le x\le b$a≤x≤b, we have $\int_a^bf(x)dx=A$∫baf(x)dx=A.
What does the notation $\int_a^bf(x)dx$∫baf(x)dx entail mathematically?
Calculating a definite integral is a three-step process:
- Calculate an antiderivative, $F\left(x\right)$F(x)
- Substitute in the upper and lower limits and subtract: $F\left(b\right)-F\left(a\right)$F(b)−F(a)
- Evaluate
Once we have calculated the anti-derivative, we use square brackets around this primitive function and place the upper and lower limits on the right bracket. This notation shows in one step both the integral and what we're getting ready to substitute into our anti-derivative.
Let's look at some examples of how this is done below.
Worked examples
Example 1
Evaluate 
Example 2
Evaluate 
Practice questions
Question 1
Calculate $\int_2^4\left(6x+5\right)dx$∫42(6x+5)dx.
Question 2
Calculate $\int_{-2}^1\left(x^2+4\right)dx$∫1−2(x2+4)dx.
Question 3
Calculate $\int_6^{11}\sqrt{x-2}dx$∫116√x−2dx.
Properties of definite integrals
Definite integrals have a number of properties that we can make use of in order to simplify and solve problems involving area and integration. These properties are summarised below.
- Equal end-points:
- Function multiplied by a constant:
- Switch end-points:
- Sum of functions:
- Split interval:
The above properties are useful in manipulating integrals to simplify a problem or solve a problem given partial information. Let's look briefly at each property:
1. End-points equal:
This property is can be shown from the definition of a definite integral:
This also concurs with the idea that the area under a point is zero.
2. Function multiplied by a constant:
Just as for differentiation, when a function is multiplied by a constant $k$k and integrated, it is is the same as multiplying the integral of the function by the constant.
3. Switch end-points:
If we reverse the interval of a definite integral, that is instead of integrating from $a$a to $b$b we integrate from $b$b to $a$a we will get the negative of the original integral. This can be shown to be true from the definition of a definite integral as follows:
4. Sum of functions:
If a definite integral is the sum or difference of two functions, then this is the same as adding or subtracting the individual integral of each function over the interval. As with differentiation this means we can integrate each part of a function separately.
5. Split interval:
This property is very useful to combine to integrals with a shared start/end point or to split a given integral into separate regions.
We can split an integral over a closed interval $\left[a,c\right]$[a,c] at the point $b$b where $b$b is within the closed interval, the sum of the integral of the parts equals the original integral.
This can be shown from the definition of a definite integral as follows:
Practice questions
Question 4
Consider the function $f\left(x\right)=6x$f(x)=6x.
Find the definite integral, $\int_4^8f\left(x\right)dx$∫84f(x)dx.
Find the definite integral, $\int_8^4f\left(x\right)dx$∫48f(x)dx.
Which property of definite integrals do parts (a) and (b) demonstrate?
$\int_a^bf\left(x\right)dx=-\int_{\editable{}}^{\editable{}}f\left(x\right)dx$∫baf(x)dx=−∫f(x)dx
Question 5
Suppose $\int_{-1}^2f\left(x\right)dx=4$∫2−1f(x)dx=4 and $\int_2^8f\left(x\right)dx=8$∫82f(x)dx=8.
Find the value of $\int_{-1}^8f\left(x\right)dx$∫8−1f(x)dx.
Find the value of $\int_8^{-1}f\left(x\right)dx$∫−18f(x)dx.
Find the value of $\int_{-1}^22f\left(x\right)dx+\int_2^83f\left(x\right)dx$∫2−12f(x)dx+∫823f(x)dx.