Processing Math: Done

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= −cosx 0− −cosx 54=−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 cos(54) is to draw the angles 0, and 54 on a unit circle and to read off the cosine value as the x-coordinate for the corresponding point on the circle.