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

1 Answer
Feb 26, 2017

Answer:

#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:
enter image source here

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#

enter image source here

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

enter image source here

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.