Consider the integral K= _ 0 ^ 4 | x- 1 2 | ,dx

MATHEMATICS · Subjective · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Consider the integral

K=\int_{0}^{4\pi} \left|\sin x-\frac{1}{2}\right|\,dx.

(i) Find all values of

x

in the interval

[0,4\pi]

at which

\sin x=\dfrac{1}{2}.

(ii) Using these points, evaluate

K

exactly by integrating over each resulting sub-interval.