Find the total area of the region trapped between y=sin x and y=cos x and the lines x=pi/4 and x=21/4. Sketch the region.
Please help! I have the graph sketched already but I don't know how to do anything else.
The area enclosed by the curves y = sin x, y = cos x and the lines `x = pi/4` and `x = 21/4` has to be determined.
From the graph it is seen that the curves y = sin x and y = cos x intersect at the points where `x = pi/4` and `x = (5*pi)/4`
The area enclosed between these curves is:
`int_(pi/4)^((5*pi)/4) sin x - cos x dx`
= `[-cos x - sin x]_(pi/4)^((5*pi)/4)`
= `-cos ((5*pi)/4) + cos (pi/4) - sin ((5*pi)/4) + sin(pi/4)`
= `1/sqrt 2 + 1/sqrt 2 + 1/sqrt 2 + 1/sqrt 2`
= `4/sqrt 2`
= `2*sqrt 2`
The enclosed area is `2*sqrt 2`