First slide

Area of bounded Regions

Question

$Evaluate \int_{0}^{2 π} [\sin x] d x, where [.] denotes the greatest integer function.$

Difficult

A

$- π$
B

$π$
C

$2 π$
D

$- 2 π$

Solution

$y = \sin x is a non-monotonic function in [0, 2 π] .$
$Hence, draw the graph of f (x) = [\sin x] .$
$From the above graph we have,$
$\begin{array}{c} \int_{0}^{2 π} [\sin x] d x = Algebraic area of the shaded region \\ \begin{aligned} = (π) (- 1) (l \times b) \\ = - π \end{aligned} \end{array}$

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