What is the total area of the shaded regions bounded by the curves y = sin x and y = cos x?

Question

Find the total area of the shaded regions bounded by the curves #y = sinx# and #y = cos x# for 0#<=##x##<=##pi/2#

2755 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-22T09:56:36

#Area = 2sqrt2-2#

In the domain, #0 <=x <= pi/4#, the sine function is less than the cosine function, therefore, the area for this first region is:

#Area_1 = int_0^(pi/4) cos(x)-sin(x)dx#

#Area_1 = (sin(x)+cos(x)|_0^(pi/4)#

#Area_1 = sin(pi/4)+cos(pi/4) - (sin(0)+cos(0))#

#Area_1 = sqrt2/2+sqrt2/2 - 0-1#

#Area_1 = sqrt2-1#

In the domain, #pi/4 < x <=pi/2 #, the cosine function is less than the sine function, therefore, the area of the second region is:

#Area_2 = int_(pi/4)^(pi/2) sin(x)-cos(x) dx #

#Area_2 = (-cos(x)-sin(x)|_(pi/4)^(pi/2)#

#Area_2 = -cos(pi/2)-sin(pi/2)+cos(pi/4)+sin(pi/4)#

#Area_2 = 0-1+sqrt2/2+sqrt2/2#

#Area_2 = sqrt2-1#

The total area is:

#Area = Area_1+Area_2#

#Area = sqrt2-1+sqrt2-1#

#Area = 2sqrt2-2#