Question

Find the exact area bounded by the curves y=sinx and y=cosx in the domain 0≤x≤2pi

Accepted Answer

Let,
EQ.1   y = cos x             EQ.2  y = sin x                              
Solve the intersection points between the two equations.
             y = y      
       sin x = cos x   
Divide both sides by cos x.
       (sin x)/(cosx) = 1        
      tan x = 1  
Refer to Unit Circle Chart or Table of Trigonometric Function for Special Angle to determine the value of x.
So at the interval 0<=x<=2pi  values of x are,
     x = pi/4 and (5pi)/4      
Then substitute these values to either EQ.1 or EQ.2.           
 y = cos x = cos (pi/4) = sqrt(2)/2          
 y = cos x = cos ((5pi)/4) = -sqrt2/2                                      
So the two equations intersect at  points (pi/4, sqrt2/2)   and  ((5pi)/4, sqrt2/2) .
The graph of the two equations is: 
(Note: The color of the graph of EQ.1 is red, while EQ.2 is purple.)
  
 
Then, use the formula for area between two curves.
A = int y_U - y_L dx
Use the x-coordinate of the intersection points as the limits of the integral.                        
A= int_(pi/4)^((5pi)/4) (sin x - cosx) dx = -cosx - sinx |_(pi/4)^((5pi)/4)
A= -cos x - sin x |_(pi/4)^((5pi)/4) =(-cos ((5pi)/4) - sin((5pi)/4)) - (-cos (pi/4) - sin (pi/4))
A = (-(-sqrt2/2 )- (-sqrt2/2)) - (-sqrt2/2 - sqrt2/2) = (sqrt2/2 + sqrt2/2) - (-sqrt2/2 - sqrt2/2)
A = sqrt2 - (-sqrt2) = sqrt2+sqrt2 = 2sqrt2
Answer: Area bounded by two curves is 2sqrt2  .

Math

Find the exact area bounded by the curves y=sinx and y=cosx in the domain 0≤x≤2pi

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