First slide

Area of bounded Regions

Question

The value of the integral $\int_{1}^{3} [x^{2} - 2 x - 2] d x$ , where $[x]$ denotes the greatest integer less or equal to x, is

Moderate

A

$- \sqrt{2} - \sqrt{3} + 1$
B

$- \sqrt{2} - \sqrt{3} - 1$
C

-4
D

-5

Solution

$\begin{array}{rcl} [x^{2} - 2 x - 2] & = & - 3 1 \leq x \leq 2 \\ = & - 2 2 \leq x \leq 1 + \sqrt{2} \\ = & - 1 1 + \sqrt{2} \leq x \leq 1 + \sqrt{3} \\ = & 0 1 + \sqrt{3} \leq x \leq 3 \end{array}$
The required integral value is
$\int_{1}^{2} - 3 d x + \int_{2}^{1 + \sqrt{2}} - 2 + \int_{1 + \sqrt{2}}^{1 + \sqrt{3}} - 1 + \int_{1 + \sqrt{3}}^{3} 0 = - \sqrt{2} - \sqrt{3} - 1$

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