Consider the following for the two (02) items that follow: Let f(x) = [x^2] , where [ ] is the greatest integer function. What is _ 2 ² f(x) , dx equal to?

Step-by-step solution

The given integral is _ 2 ² [x^2] , dx . As x varies from 2 to 2 , x^2 varies from 2 to 4 . The greatest integer function [x^2] changes its value when x^2 is an integer. The integer values of x^2 in this interval are 2, 3, and 4 , which correspond to x = 2 , 3 , and 2 . We split the integral at x = 3 : For x ( 2 , 3 ) , 2 For x ( 3 , 2) , 3 The integral can be written as: _ 2 ² [x^2] , dx = _ 2 ^ 3 2 , dx + _ 3 ² 3 , dx Evaluating the integrals: _ 2 ^ 3 2 , dx = 2[x]_ 2 ^ 3 = 2( 3 - 2 ) = 2 3 - 2 2 _ 3 ² 3 , dx = 3