Integration Formula:
Integrate each term:
Altogether this becomes:
Your limits are 1 and 2. Substitute both numbers into the equation and minus them from each other.
x = 2:
-
x = 1:
[/tex]
x = 2:
-
x = 1:
=
-1.212317928
Now convert all the answers to integers and see which one matches -1.212317928:
Because it's a negative number, we know it can only be A or B.
(A) = -1.212317928
(B) = -1.19047619
The answer is A:
The cosine of an angle is the x-coordinate of the point where its terminal ray intersects the unit circle. So, we can draw a line at x=-1/2 and see where it intersects the unit circle. That will tell us possible values of θ/2.
We find that vertical line intersects the unit circle at points where the rays make an angle of ±120° with the positive x-axis. If you consider only positive angles, these angles are 120° = 2π/3 radians, or 240° = 4π/3 radians. Since these are values of θ/2, the corresponding values of θ are double these values.
a) The cosine values repeat every 2π, so the general form of the smallest angle will be
... θ = 2(2π/3 + 2kπ) = 4π/3 + 4kπ
b) Similarly, the values repeat for the larger angle every 2π, so the general form of that is
... θ = 2(4π/3 + 2kπ) = 8π/3 + 4kπ
c) Using these expressions with k=0, 1, 2, we get
... θ = {4π/3, 8π/3, 16π/3, 20π/3, 28π/3, 32π/3}
