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Mathematics
Mathematics
Problem
Consider the integral
K
=
∫
0
4
π
∣
sin
x
−
1
2
∣
d
x
.
K=\int_{0}^{4\pi} \left|\sin x-\frac{1}{2}\right|\,dx.
K
=
∫
0
4
π
sin
x
−
2
1
d
x
.
(i) Find all values of
x
x
x
in the interval
[
0
,
4
π
]
[0,4\pi]
[
0
,
4
π
]
at which
sin
x
=
1
2
.
\sin x=\dfrac{1}{2}.
sin
x
=
2
1
.
(ii) Using these points, evaluate
K
K
K
exactly by integrating over each resulting sub-interval.
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