The absolute value forces the answers to definite
integrals to be positive on the interval [0,pi/2]. Therefore the integral needs to be
split up. Looking at the graph of sin x - sqrt 3 (cos x), from [0,pi/3] the y-values
are negative. From [pi/3, pi/2] the y-values are
positive.
The
definite integral breaks into two.
class="AM"> from [0,pi/3]
plus
sin(x)-sqrt(3)cos(x)dx from [pi/3, pi/2]
The
integral is equal to -cos(x)-sqrt(3)*sin(x)
Putting in the
bounds:
-[-cos(pi/3)-sqrt(3)sin(pi/3)-(-cos(0)-sqrt(3)sin(0))]+
[-cos(pi/2)-sqrt(3)sin(pi/2)-(-cos(pi/3)-sqrt(3)sin(pi/3))]
=
-[-1/2-3/2-(-1-0)] + [-0-sqrt(3)-(-1/2-3/2)]
=
-[-1]+[2-sqrt(3)] =
3-sqrt(3)
The exact answer is 3-sqrt(3) which is
approximately 1.268.
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