Recall that for two continuous curves $y = f(x)$ and $y = g(x)$, if f(x) ≥ g(x) for a ≤ x ≤ b, then it follows that the area trapped between these two curves is:

(1)Similarly, for two continuous curves $x = f(y)$ and $x = g(y)$, if f(y) ≥ g(y) for c ≤ d, then it follows that the area trapped between these two curves is:

(2)We will now apply this technique in the following examples.

## Example 1

**Determine the area trapped between the curves $f(x) = 5x - x^2$ and $g(x) = x$.**

We note that these two graphs intersect each other at x = 0 and x = 4. Hence our limits of integration will be 0 and 4. Furthermore, we note that f(x) ≥ g(x) for 0 ≤ x ≤ 4 as shown when graphed below:

We now want to determine the area of the shaded region, which we can do by using the formula at the top of the page:

(3)## Example 2

**Calculate the area trapped between the curves $f(x) = x$, $g(x) = x^2$.**

We first note that f(x) and g(x) intersect at x = 0 and x = 1. Hence our limits of integration are 0 and 1. Furthermore, we note that f(x) ≥ g(x) for 0 ≤ x ≤ 1 as shown when the graphs are drawn:

Hence it follows that:

(4)## Example 3

**Calculate the area trapped between the curves $f(x) = 1/x$, $g(x) = 1/x^2$, and $x = 2$.**

We note that the curves f(x) and g(x) intersect at x = 1. We were also given the upper limit of integration. Hence our limits of integration are 1 and 2. Furthermore, we note that f(x) ≥ g(x) for 1 ≤ x ≤ 2 (which we can verify by testing any value of x on this interval). The area we are trying to calculate is shown in the following graph:

All we need to do is evaluate the following integral:

(5)## Example 4

**Calculate the are trapped between the curves $f(y) = 2y^2$ and $g(y) = 4 + y^2$.**

We first note that these two curves intersect when y = -2 and y = 2. Hence our limits of integration are -2 and 2. Furthermore, we note that g(y) ≥ f(y) for -2 ≤ y ≤ 2. Hence we can evaluate this integral as follows:

(6)