In this section we learn how to calculate the area enclosed between two curves, using definite integrals.
The typical type of scenario we'll be interested in is shown here. We'll often be required to calculate the area enclosed (or stuck between) two curve \(y = f(x)\) and \(y = g(x)\).
The method is explained in the following series of tutorials.
In this first tutorial we learn the formula for calculating the area enclosed beyween two curves, as well as highlight the three steps we should always follow when using it.
Given two curves \(y=f(x)\) and \(y=g(x)\), which interesect at two points, with \(x\)-coordinates: \[x = a \quad \text{and} \quad x = b\] such that \(f(x) \geq g(x)\), for \(a\leq x \leq b \), which means that \(y=f(x)\) is higher than (or above) \(y = g(x)\) over the interval \(a\leq x \leq b\).
The area enclosed by these two curves can be calculated using the formula: \[\text{Enclosed Area} = \int_a^b \begin{pmatrix}f(x)-g(x)\end{pmatrix}dx\]
In this second tutorial we look at a worked example, in which we're given the coordinates of the points of intersection of the two curves. This should give us a good idea of how the formula "works".
The three-step method we've just seen is summarized here (make a note of it).
Using the three-step method, we saw above, try working through each of the following questions.
As good pratice, try solving each of these with and without using a graphical calculator.