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Aus Online Mathematik Brückenkurs 2

If we draw the curve y=sinx, we see that the curve lies above the x -axis as far as x= and then lies under the x -axis.

The area of the region between x=0 and x= can therefore be written as

0sinxdx 

whilst the area of the remaining region under the x -axis is equal to

−54sinxdx 

(note the minus sign in front of the integral).

The total area becomes

0sinxdx−54sinxdx=−cosx0−−cosx54=−cos−−cos0−−cos45−−cos=−−1−−1−−−12−−−1=1+1−12+1=3−12

NOTE: a simple way to obtain the values of cos0, cos and cos45 is to draw the angles 0, and 45 on a unit circle and to read off the cosine value as the x -coordinate for the corresponding point on the circle.