First slide

Evaluation of definite integrals

Question

$Evaluate \int_{0}^{100} (x - [x]) d x (where [] re- presents the greatest integer function).$

Moderate

A

50
B

100
C

200
D

150

Solution

$\begin{aligned} x - [x] = {x} has period 1 . \\ Thus, \begin{matrix} \int_{0}^{100} (x - [x]) d x & = 100 \int_{0}^{1} {x} d x \\ = 100 \int_{0}^{1} x d x (∵ for x \in (0, 1), (x} = x) \\ = \frac{100}{2} {[x^{2}]}_{0}^{1} = 50 \end{matrix} \end{aligned}$

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