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The area bounded by the curve y = x[x], the x – axis and the ordinates x = 1 and x = -1 is given by
- a)0
- b)1/2
- c)2/3
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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Understanding the Problem
To find the area bounded by the curve y = x[x], the x-axis, and the vertical lines x = -1 and x = 1, we first need to determine the behavior of the function y = x[x].
Analyzing the Function
- The function y = x[x] involves the greatest integer function [x], which takes the largest integer less than or equal to x.
- For x in the interval [-1, 0), [x] = -1, resulting in y = x * (-1) = -x.
- For x in the interval [0, 1), [x] = 0, resulting in y = x * 0 = 0.
This means we can break the area calculation into two parts:
Calculating Areas
1. For x in [-1, 0):
- The function simplifies to y = -x.
- The area under this curve from x = -1 to x = 0 is given by the integral:
Area = ∫[-1 to 0] (-x) dx = [-(x^2)/2] from -1 to 0 = [0 - (-1/2)] = 1/2.
2. For x in [0, 1):
- The function simplifies to y = 0.
- The area under this curve from x = 0 to x = 1 is 0.
Total Area Calculation
- The total area bounded by the curve, the x-axis, and the ordinates is simply the area calculated from the interval [-1, 0], which is:
Total Area = Area from [-1, 0] + Area from [0, 1] = 1/2 + 0 = 1/2.
Conclusion
Thus, the area bounded by the curve y = x[x], the x-axis, and the ordinates x = -1 and x = 1 is indeed 1/2, confirming that the correct answer is option 'B'.
To find the area bounded by the curve y = x[x], the x-axis, and the vertical lines x = -1 and x = 1, we first need to determine the behavior of the function y = x[x].
Analyzing the Function
- The function y = x[x] involves the greatest integer function [x], which takes the largest integer less than or equal to x.
- For x in the interval [-1, 0), [x] = -1, resulting in y = x * (-1) = -x.
- For x in the interval [0, 1), [x] = 0, resulting in y = x * 0 = 0.
This means we can break the area calculation into two parts:
Calculating Areas
1. For x in [-1, 0):
- The function simplifies to y = -x.
- The area under this curve from x = -1 to x = 0 is given by the integral:
Area = ∫[-1 to 0] (-x) dx = [-(x^2)/2] from -1 to 0 = [0 - (-1/2)] = 1/2.
2. For x in [0, 1):
- The function simplifies to y = 0.
- The area under this curve from x = 0 to x = 1 is 0.
Total Area Calculation
- The total area bounded by the curve, the x-axis, and the ordinates is simply the area calculated from the interval [-1, 0], which is:
Total Area = Area from [-1, 0] + Area from [0, 1] = 1/2 + 0 = 1/2.
Conclusion
Thus, the area bounded by the curve y = x[x], the x-axis, and the ordinates x = -1 and x = 1 is indeed 1/2, confirming that the correct answer is option 'B'.
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