GATE ME 2020 Set 1 — Question 5
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Engineering Mathematics → Single Variable Calculus → Definite & Improper Integrals
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Question
Define as the greatest integer less than or equal to , for each . If , then area under for is ______.
Correct answer
(D) 6
Solution
The problem asks for the area under the curve of the function for in the interval . The function represents the greatest integer less than or equal to .The area can be calculated by integrating the function from to .Area The greatest integer function is a step function. We can break the integral into intervals where the value of is constant:
1.For , .
2.For , .
3.For , .
So, we can split the integral as follows:Area Now, substitute the constant values of in each interval:Area Evaluate each integral:Total Area .Alternatively, we can visualize this as the sum of the areas of three rectangles:- Rectangle 1: width = , height = . Area = .
- Rectangle 2: width = , height = . Area = .
- Rectangle 3: width = , height = . Area = .
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