# What is the area of the region in the first quadrant enclosed by the graphs of y = cosx, y= x, and the y -axis?

Feb 26, 2017

$A = 0.401 {u}^{2}$

#### Explanation:

To find the area between two curves, find the integral of the difference between the two functions over the desired interval.

That is a mouthful, so it is probably best to explain using a graph: We are looking for the purple area between these two curves. First, let's figure out where $y = \cos \left(x\right)$ and $y = x$ intersect.

Unfortunately, there is not an easy way to find the intersection of these two functions by hand. Using a graphing calculator, it can be seen that $y = \cos \left(x\right)$ and $y = x$ intersect at (0.739, 0.739).

Now, let's integrate:

Function 1:
${\int}_{0}^{0.739} \cos \left(x\right) \mathrm{dx} = \sin \left(0.739\right) - \sin \left(0\right) = 0.674 - 0 = 0.674$ Function 2:
${\int}_{0}^{0.739} x \mathrm{dx} = {\left(0.739\right)}^{2} / 2 - {\left(0\right)}^{2} / 2 = 0.273$ Combining the two, we get that the area between $y = \cos \left(x\right)$ and $y = x$ bounded by the y-axis is:

$A = 0.674 - 0.273 = 0.401$

Graphically, we can see this as what is shown at the beginning.