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

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lemjay eNotes educator| Certified Educator


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`  .