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...

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.

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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`

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