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Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of #y=sin(x)# and #y=cos(x)#, how do you find the area of R?

Answer 1

Ezekiel Alexander

#sqrt(2)-1#

#R =int_0^(pi/4)(cos(x)-sin(x))dx = (sin(pi/4)+cos(pi/4))-(sin(0)+cos(0)) = 2sqrt(2)/2-1=sqrt(2)-1#

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Answer 2

Emily Ammerman

To find the area of region ( R ), calculate the definite integral of the absolute difference between ( \sin(x) ) and ( \cos(x) ) from ( x = 0 ) to the point of intersection between ( \sin(x) ) and ( \cos(x) ) in the first quadrant. This point of intersection can be found by setting ( \sin(x) = \cos(x) ) and solving for ( x ). Then, integrate ( |\sin(x) - \cos(x)| ) from ( x = 0 ) to the found point of intersection. This integral represents the area of region ( R ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.