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For which of the following values of m , is the area of the region bounded by the curve y = x - x2 and the line y = mx equal to 9/2 ?
- a)-2
- b)2
- c)-4
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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For which of the following values of m , is the area of the region bou...
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Understanding the Problem:
The problem asks us to find the value of 'm' for which the area of the region bounded by the curve y = x - x^2 and the line y = mx is equal to 9/2.
Approach to Solve:
To find the area of the region, we need to first determine the points of intersection between the curve and the line. Then, we will calculate the area bounded by these two curves using definite integrals.
Finding Points of Intersection:
1. Set the equation of the curve and the line equal to each other:
x - x^2 = mx
Rearrange the equation:
x^2 + (m-1)x = 0
2. For intersection, the discriminant of the quadratic equation should be zero:
(m-1)^2 - 4(1)(0) = 0
m^2 - 2m + 1 = 0
(m - 1)^2 = 0
m = 1
Thus, the line y = mx intersects the curve y = x - x^2 at m = 1.
Calculating Area:
1. To find the area of the region bounded by the curve and the line, we will set up the definite integral:
Area = ∫[a, b] (mx - x - x^2) dx
where 'a' and 'b' are the x-values of intersection points.
2. Substituting m = 1 and integrating the function, we get:
Area = ∫[0, 1] (x - x - x^2) dx
= ∫[0, 1] (-x^2) dx
= [-x^3/3] [0, 1]
= -1/3
Since the area is -1/3 ≠ 9/2, the correct answer is none of these.
The problem asks us to find the value of 'm' for which the area of the region bounded by the curve y = x - x^2 and the line y = mx is equal to 9/2.
Approach to Solve:
To find the area of the region, we need to first determine the points of intersection between the curve and the line. Then, we will calculate the area bounded by these two curves using definite integrals.
Finding Points of Intersection:
1. Set the equation of the curve and the line equal to each other:
x - x^2 = mx
Rearrange the equation:
x^2 + (m-1)x = 0
2. For intersection, the discriminant of the quadratic equation should be zero:
(m-1)^2 - 4(1)(0) = 0
m^2 - 2m + 1 = 0
(m - 1)^2 = 0
m = 1
Thus, the line y = mx intersects the curve y = x - x^2 at m = 1.
Calculating Area:
1. To find the area of the region bounded by the curve and the line, we will set up the definite integral:
Area = ∫[a, b] (mx - x - x^2) dx
where 'a' and 'b' are the x-values of intersection points.
2. Substituting m = 1 and integrating the function, we get:
Area = ∫[0, 1] (x - x - x^2) dx
= ∫[0, 1] (-x^2) dx
= [-x^3/3] [0, 1]
= -1/3
Since the area is -1/3 ≠ 9/2, the correct answer is none of these.
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For which of the following values of m , is the area of the region bounded by the curve y = x -x2and the line y = mx equal to9/2?a)-2b)2c)-4d)none of theseCorrect answer is option 'A'. Can you explain this answer?
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