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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 equals 9/2?
- a)– 4
- b)– 2
- c)2
- d)4
Correct answer is option 'B,D'. Can you explain this answer?
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For which of the following values of m, is the area of the region boun...
The two curves meet at
or (1 - m)3 = 27 ,
∴ m = -2
But if m >1 then 1– m is – ive, then 
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For which of the following values of m, is the area of the region boun...
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For which of the following values of m, is the area of the region boun...
The equation y = x^2 represents a parabola that opens upward, with the vertex at the origin (0,0).
To find the values of m that give the desired area, we need to find the x-values where the parabola intersects the line y = m.
Setting x^2 = m, we can solve for x:
x^2 = m
x = ±√m
The parabola intersects the line y = m at the x-values ±√m.
The area of the region between the curve and the line is given by the integral of the difference between the two functions over the interval from -√m to √m:
Area = ∫(x^2 - m) dx
Evaluating this integral gives:
Area = (1/3)x^3 - mx | from -√m to √m
= (1/3)(√m)^3 - m√m - [(1/3)(-√m)^3 - m(-√m)]
= (1/3)m√m - m√m + (1/3)m√m - m√m
= (2/3)m√m - 2m√m
We want the area to be equal to 8, so we set this expression equal to 8 and solve for m:
(2/3)m√m - 2m√m = 8
(2/3)m√m = 8 + 2m√m
(2/3)m = 8 + 2m
2m = 24 + 6m
4m = 24
m = 6
Therefore, the value of m that gives an area of 8 is m = 6.
To find the values of m that give the desired area, we need to find the x-values where the parabola intersects the line y = m.
Setting x^2 = m, we can solve for x:
x^2 = m
x = ±√m
The parabola intersects the line y = m at the x-values ±√m.
The area of the region between the curve and the line is given by the integral of the difference between the two functions over the interval from -√m to √m:
Area = ∫(x^2 - m) dx
Evaluating this integral gives:
Area = (1/3)x^3 - mx | from -√m to √m
= (1/3)(√m)^3 - m√m - [(1/3)(-√m)^3 - m(-√m)]
= (1/3)m√m - m√m + (1/3)m√m - m√m
= (2/3)m√m - 2m√m
We want the area to be equal to 8, so we set this expression equal to 8 and solve for m:
(2/3)m√m - 2m√m = 8
(2/3)m√m = 8 + 2m√m
(2/3)m = 8 + 2m
2m = 24 + 6m
4m = 24
m = 6
Therefore, the value of m that gives an area of 8 is m = 6.
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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 equals 9/2?a)– 4b)– 2c)2d)4Correct answer is option 'B,D'. Can you explain this answer?
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