Question

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

Answer 1

`A = 0.401u^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(x)` 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(x)` and `y=x` intersect at (0.739, 0.739).

Now, let's integrate:

Function 1:
`int_0^0.739 cos(x)dx = sin(0.739) - sin(0) = 0.674-0=0.674`

Function 2:
`int_0^0.739 xdx = (0.739)^2/2 - (0)^2/2=0.273`

Combining the two, we get that the area between `y=cos(x)` 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.