Problem 77 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int_{-2}^{2}\left[x^{2}\right] d x\) is equal to (A) \(10-2 \sqrt{3}-2 \sqrt{2}\) (B) \(10+2 \sqrt{3}-2 \sqrt{2}\) (C) \(10-2 \sqrt{3}+2 \sqrt{2}\) (D) None of these

Step-by-Step Solution

Answer

The integral evaluates to 2, which is none of the provided options (D).

Step 1: Understanding the Problem

We are asked to evaluate the integral \( \int_{-2}^{2} \lfloor x^2 \rfloor \, dx \), where \( \lfloor x^2 \rfloor \) is the floor function of \( x^2 \). This function rounds down \( x^2 \) to the nearest integer.

Step 2: Identify the Intervals for the Floor Function

To compute the integral, determine where \( x^2 \) takes integer values within the interval \([-2, 2]\). These critical points are at \( x = -2, -1, 0, 1, 2 \) where \( x^2 = 0, 1, 4\). Set intervals based on these points: \([-2, -1)\), \([-1, 0)\), \([0, 1)\), and \([1, 2)\).

Step 3: Evaluate the Integral on Each Interval

Since the floor function allows us to treat each segment as a constant function, integrate separately:1. On \([-2, -1)\), \( x^2 \) ranges from 1 to 4, so \( \lfloor x^2 \rfloor = 1 \).2. On \([-1, 0)\), \( x^2 \) ranges from 0 to 1, so \( \lfloor x^2 \rfloor = 0 \).3. On \([0, 1)\), \( x^2 \) ranges from 0 to 1, so \( \lfloor x^2 \rfloor = 0 \).4. On \([1, 2)\), \( x^2 \) ranges from 1 to 4, so \( \lfloor x^2 \rfloor = 1 \).

Step 4: Compute the Definite Integrals

Now compute each individual integral:- \( \int_{-2}^{-1} 1 \, dx = [x]_{-2}^{-1} = (-1) - (-2) = 1 \).- \( \int_{-1}^{0} 0 \, dx = [x]_{-1}^{0} = 0 - (-1) = 0 \).- \( \int_{0}^{1} 0 \, dx = [x]_{0}^{1} = 1 - 0 = 0 \).- \( \int_{1}^{2} 1 \, dx = [x]_{1}^{2} = 2 - 1 = 1 \).

Step 5: Sum Up the Results

Add up the results from each interval: \( 1 + 0 + 0 + 1 = 2 \). Hence, the integral evaluates to 2.

Key Concepts

Floor Function

The floor function is a mathematical function that takes a real number and rounds it down to the nearest integer less than or equal to that number. It is often represented by the symbol \( \lfloor x \rfloor \) and is also known as the greatest integer function. This function is essential when dealing with piecewise functions, especially in integration.

Key Characteristics of the Floor Function:

• The floor of a positive non-integer number is the greatest integer less than the number. For example, \( \lfloor 3.75 \rfloor = 3 \).
• The floor of a negative non-integer number behaves similarly. For example, \( \lfloor -2.5 \rfloor = -3 \) (rounding down).
• When you apply the floor function to an integer, it remains unchanged. For example, \( \lfloor 5 \rfloor = 5 \).

Understanding the floor function is crucial for correctly interpreting mathematical problems involving piecewise constant functions, like in this exercise.

Piecewise Integration

Piecewise integration involves breaking up an integral into sections where the function has distinct definitions. This approach is necessary when the function being integrated is not continuous over the interval or when the analytical form of the function changes across different subintervals.

Steps in Piecewise Integration:

• Identify the points where the function changes its definition. In this exercise, the critical points were identified as \(-2, -1, 0, 1,\) and \(2\), based on where \(x^2\) took integer values.
• Define separate integrals for each interval between these points. Hence, in the provided solution, the integrals were divided into the four segments: \([-2, -1)\), \([-1, 0)\), \([0, 1)\), and \([1, 2)\).
• Evaluate each integral separately, using the specific formula of the function for each interval. Since the floor function makes the function constant over each segment, computation is simplified.

Piecewise integration is particularly useful when dealing with step functions or functions with breaks or discontinuities, ensuring the correct evaluation of each section.

Calculus Step-by-Step Solution

Solving calculus problems step by step allows for a deeper understanding and better clarity of how to solve an integral. This process involves breaking down the integral into smaller, manageable pieces, and working through them systematically.

Step-by-Step Approach in the Given Solution:

• Understanding the Problem: Initially, interpret the expression and the role of each part, such as recognizing how the floor function modifies the behavior of \(x^2\).
• Identify Intervals: Determine where within the given domain the input of the floor function would change its output, which was identified as points where \(x^2\) becomes an integer.
• Segment Analysis: Calculate the integral over each of these intervals separately, acknowledging that the floor function within each piece improves consistency and simplicity over the integrals.
• Summing Up: Compile results from all the integrated segments to find the final solution.

This meticulous method ensures all parts of a problem are addressed and reduces complexity by managing each piece individually. Step-by-step solutions contribute significantly towards building a strong foundation in solving more complex calculus problems.