Calculus: Early Transcendentals, Chapter 6, 6.1, Section 6.1, Problem 10

Here is the sketch of the two given functions.
The y = sin(x) is plotted with a red color while y = (2x)/pi is plotted with a blue color.
As shown the graph, the two graphs intersect at the following points (approximately):
---> (-1.57 -1)
---> (0,0)
---> (1.57, 1).
The x-values from the intersection points will be used as the limits of integration or boundary values of x for each bounded region.

Using integration with respect to x, we follow the formula for the "Area between Two Curves" as:
A = int_a^bf(x)-g(x)dx
such that f(x)gt=g(x) for interval [a,b]
Or A =int_a^b[y_(above)-y_(below)]dx


Please see the attached file: "graph" to view how a typical approximating rectangle (sky blue in color) is used when using integration with respect to x. In the attached file, the width =dx and the height =f(x) such that
f(x)= y_(above) - y_(below).
To find the area of a bounded region, we will solve each bounded region with two separate integral then find the sum for the total bounded area/region.
In the first bounded region (yellow in shade), we have the y_(above)= (2x)/pi
and y_(below) = sin(x) with limits of integration from x =-1.57 to x=0.
A_1= int_-1.57^0[(2x)/pi -sin(x)]dx
=[cos(x)+x^2/pi]|_(-1.57)^0
= [cos(0)+0^2/pi]-[cos(-1.57)+(-1.57)^2/pi]
= [1+0] - [ 0.00079633+0.7846020385]
= 1 - 0.7853983685
~~ 0.2146
In the second bounded region (pink in shade), we have the y_(above)=sin(x)
and y_(below) = (2x)/pi with limits of integration from x =0 to x=1.57.
A_2 =int_0^1.57[sin(x)-(2x)/pi]dx
=[-cos(x) - x^2/pi]|_0^(1.57)
=[-cos(1.57)-(1.57)^2/pi]-[-cos(0)-0^2/pi]
= [ -0.00079633-0.7846020385] -[-1-0]
= - 0.7853983685+1
~~ 0.2146


Notice that they are symmetrical about the origin.
We can multiply A_1 by 2 since the two bounded area is the same.
Total Area=
= 0.2146 +0.2146
=0.4292
In the first bounded region (yellow in shade), we have the y_(above)= (2x)/pi
and y_(below) = sin(x) with limits of integration from x =-1.57 to x=1.
In the first bounded region (yellow in shade), we have the y_(above)= (2x)/pi
and y_(below) = sin(x) with limits of integration from x =-1.57 to x=1.