In this section we learn how to calculate the area enlcosed by a curve \(y=f(x)\) between the \(x\)-axis and two values of \(x\), typically referred to as \(x = a\) and \(x = b\).
The type of area we'll know how to calculate is illustrated in the following sketch:
In each sketch, the area shaded in blue represents the area we'll soon know how to calculate.
We can see that the area corresponds to the area enclosed by the curve and the \(x\)-axis, between \(x=a\) and \(x=b\).
We start by learning the method, the formula, we'll then watch a tutorial and finally we'll work our way through a series of exercises.
The area encosed by a curve between the \(x\)-axis, \(x=a\) and \(x=b\) can be calculated with the formula: \[\text{Area} = \int_a^b \begin{vmatrix}f(x) \end{vmatrix}dx\] Where: \[\begin{vmatrix} f(x) \end{vmatrix} = \begin{cases} -f(x) \quad \text{when} \quad f(x) < 0 \\ f(x) \quad \text{when} \quad f(x) \geq 0 \end{cases} \] This formula is better explained in the tutorial, below, make sure to watch it now.
In the following tutorial we review the method for calculating the area enclosed by a curve and the \(x\)-axis, between two values of \(x\).
For further explanation of the formula for the area enclosed by a curve, click on the button below.
Otherwise go straight to the exercises below.
Using the method just shown, calculate each of the following areas:
On top of knowing how to calculate by hand, it is very important that we know how to calculate the area enclosed by a curve using a Graphical Calculator.
Indeed, this is particularly important, as some integrals cannot easily be found by hand.
Important: when calculating areas under curves with a calculator make sure to always use the absolute value of the function.
The method for doing this is shown here, for the TI NSPire CX.
Watch the following tutorial to learn more.