find the area of the shaed region
y= cosx
y=sinx
the intesection between them is pi/4 and 5pi/4
To find the area of the shaded region between the two curves y = cosx and y = sinx, we first need to determine the points of intersection.
y = cosx and y = sinx intersect at x = pi/4 and x = 5pi/4.
Next, we find the area between the curves by subtracting the curve y = sinx from the curve y = cosx from x = pi/4 to x = 5pi/4.
To find the area, we integrate the function f(x) = cosx - sinx from x = pi/4 to x = 5pi/4:
∫[pi/4, 5pi/4] (cosx - sinx) dx
= [sinx + cosx] ∣[pi/4, 5pi/4]
= (sin(5pi/4) + cos(5pi/4)) - (sin(pi/4) + cos(pi/4))
= (-sqrt(2)/2 - sqrt(2)/2) - (1/sqrt(2) + 1/sqrt(2))
= -sqrt(2) - 1
Therefore, the area of the shaded region between the curves y = cosx and y = sinx is √2 + 1.