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The area bounded by the curves y = x2, y = [x+1], x < 1 and the y - axis, where[.] denotes the greatest integer not exceeding x, is
- a)2/3
- b)1/3
- c)1
- d)2
Correct answer is option 'A'. Can you explain this answer?
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The area bounded by the curves y = x2, y = [x+1], x < 1 and the y ...
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The area bounded by the curves y = x2, y = [x+1], x < 1 and the y ...
To find the area bounded by the given curves, we need to determine the points of intersection between the curves and the boundaries of the region.
1. Finding the point of intersection:
Let's first find the point of intersection between the curves y = x^2 and y = [x].
For y = x^2, we have:
x^2 = [x]
To find the intersection points, we need to consider two cases:
Case 1: When x is an integer
In this case, [x] will be equal to x. So, the equation becomes:
x^2 = x
x^2 - x = x(x - 1) = 0
This gives us two solutions: x = 0 and x = 1.
Case 2: When x is not an integer
In this case, [x] will be equal to the greatest integer less than or equal to x. So, the equation becomes:
x^2 = x - 1
x^2 - x + 1 = 0
Using the quadratic formula, we can solve for x, but we can see that the discriminant (b^2 - 4ac) is negative, which means there are no real solutions.
Therefore, the only points of intersection are (0, 0) and (1, 1).
2. Determining the boundaries of the region:
The given boundaries are y = [x], y = 0, and x = 1.
Since y = [x] represents the greatest integer less than or equal to x, it means that y will be an integer for all values of x between two consecutive integers. Therefore, the boundary y = [x] is equivalent to y = 0 for x < 1="" and="" y="1" for="" 1="" ≤="" x="" />< />
3. Calculating the area:
The area bounded by the curves can be divided into two regions:
Region 1: Bounded by y = x^2, y = 0, and x = 1
This region is a right-angled triangle with base 1 and height 1 (the point of intersection (1, 1)). Therefore, the area of this region is 1/2 * base * height = 1/2 * 1 * 1 = 1/2.
Region 2: Bounded by y = x^2, y = 1, and x = 2
This region is a trapezium with bases of lengths 2 and 1 (the points of intersection (0, 0) and (1, 1)). The height of the trapezium is 1 (the difference between the y-coordinates of the points of intersection). Therefore, the area of this region is 1/2 * (base1 + base2) * height = 1/2 * (2 + 1) * 1 = 3/2.
Total area = Area of Region 1 + Area of Region 2 = 1/2 + 3/2 = 2/3.
Therefore, the correct answer is option A, 2/3.
1. Finding the point of intersection:
Let's first find the point of intersection between the curves y = x^2 and y = [x].
For y = x^2, we have:
x^2 = [x]
To find the intersection points, we need to consider two cases:
Case 1: When x is an integer
In this case, [x] will be equal to x. So, the equation becomes:
x^2 = x
x^2 - x = x(x - 1) = 0
This gives us two solutions: x = 0 and x = 1.
Case 2: When x is not an integer
In this case, [x] will be equal to the greatest integer less than or equal to x. So, the equation becomes:
x^2 = x - 1
x^2 - x + 1 = 0
Using the quadratic formula, we can solve for x, but we can see that the discriminant (b^2 - 4ac) is negative, which means there are no real solutions.
Therefore, the only points of intersection are (0, 0) and (1, 1).
2. Determining the boundaries of the region:
The given boundaries are y = [x], y = 0, and x = 1.
Since y = [x] represents the greatest integer less than or equal to x, it means that y will be an integer for all values of x between two consecutive integers. Therefore, the boundary y = [x] is equivalent to y = 0 for x < 1="" and="" y="1" for="" 1="" ≤="" x="" />< />
3. Calculating the area:
The area bounded by the curves can be divided into two regions:
Region 1: Bounded by y = x^2, y = 0, and x = 1
This region is a right-angled triangle with base 1 and height 1 (the point of intersection (1, 1)). Therefore, the area of this region is 1/2 * base * height = 1/2 * 1 * 1 = 1/2.
Region 2: Bounded by y = x^2, y = 1, and x = 2
This region is a trapezium with bases of lengths 2 and 1 (the points of intersection (0, 0) and (1, 1)). The height of the trapezium is 1 (the difference between the y-coordinates of the points of intersection). Therefore, the area of this region is 1/2 * (base1 + base2) * height = 1/2 * (2 + 1) * 1 = 3/2.
Total area = Area of Region 1 + Area of Region 2 = 1/2 + 3/2 = 2/3.
Therefore, the correct answer is option A, 2/3.
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The area bounded by the curves y = x2, y = [x+1], x < 1 and the y -axis, where[.] denotes the greatest integer not exceeding x, isa)2/3b)1/3c)1d)2Correct answer is option 'A'. Can you explain this answer?
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