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The area bounded by the curve y = 2x - x2 and the line x + y = 0 is
- a)9/2 sq. units
- b)35/6 sq. units
- c)19/6 sq. units
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The area bounded by the curve y = 2x -x2 and the line x + y = 0 isa)9/...
The equation y = 2x − x2 i.e. y – 1 = - (x - 1)2 represents a downward parabola with vertex at (1, 1) which meets x – axis where y = 0 .i .e . where x = 0 , 2. Also , the line y = - x meets this parabola where – x = 2x − x2 i.e. where x = 0 , 3.
Therefore , required area is :
Therefore , required area is :
Most Upvoted Answer
The area bounded by the curve y = 2x -x2 and the line x + y = 0 isa)9/...
To find the area bounded by the curve y = 2x - x^2 and the line x = 0, we need to find the points of intersection between the curve and the line.
- Find the points of intersection:
Setting x = 0 in the equation of the curve, we get y = 0. So, the point of intersection is (0, 0).
- Determine the limits of integration:
The curve intersects the x-axis at x = 0 and x = 2. Therefore, the limits of integration will be from x = 0 to x = 2.
- Set up the integral:
The area bounded by the curve and the line is given by the definite integral of the difference between the curve and the line:
A = ∫[0,2] (2x - x^2) dx
- Evaluate the integral:
Integrating the expression, we get:
A = [x^2 - (x^3)/3] from 0 to 2
= [(2)^2 - ((2)^3)/3] - [(0)^2 - ((0)^3)/3]
= [4 - (8/3)] - [0 - 0]
= [12/3 - 8/3]
= 4/3
Therefore, the area bounded by the curve y = 2x - x^2 and the line x = 0 is 4/3 square units.
Since none of the given options matches the correct answer, it seems there might be an error in the options provided. However, based on the calculation, the correct answer is not option 'A'.
- Find the points of intersection:
Setting x = 0 in the equation of the curve, we get y = 0. So, the point of intersection is (0, 0).
- Determine the limits of integration:
The curve intersects the x-axis at x = 0 and x = 2. Therefore, the limits of integration will be from x = 0 to x = 2.
- Set up the integral:
The area bounded by the curve and the line is given by the definite integral of the difference between the curve and the line:
A = ∫[0,2] (2x - x^2) dx
- Evaluate the integral:
Integrating the expression, we get:
A = [x^2 - (x^3)/3] from 0 to 2
= [(2)^2 - ((2)^3)/3] - [(0)^2 - ((0)^3)/3]
= [4 - (8/3)] - [0 - 0]
= [12/3 - 8/3]
= 4/3
Therefore, the area bounded by the curve y = 2x - x^2 and the line x = 0 is 4/3 square units.
Since none of the given options matches the correct answer, it seems there might be an error in the options provided. However, based on the calculation, the correct answer is not option 'A'.
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