`int_(pi/6)^(2pi)(cos(x))dx` Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.

Expert Answers
tiburtius eNotes educator| Certified Educator

The integral actually represents difference between red area and green area shown in the image below i.e.

`int_(pi/6)^(2pi)cos(x)dx=P_1-P_2+P_3`

`int_(pi/6)^(2pi)cos(x)dx=sin(x)|_(pi/6)^(2pi)=sin(2pi)-sin(pi/6)=0-1/2=-1/2`

In other words the area below the graph is greater by `1/2` than the area above the graph.

`P_1-P_2+P_3=-1/2` 

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