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The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is :
- a)15/4
- b)15/2
- c)21/2
- d)17/4
Correct answer is option 'B'. Can you explain this answer?
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The area (in sq. units) of the region bounded by the parabola, y = x2 ...
Check the intersection points of the parabola and the lines and calculate its area.
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The area (in sq. units) of the region bounded by the parabola, y = x2 ...
To find the area bounded by the parabola y = x^2 - 2 and the lines y = x - 1, x = 0, and x = 3, we need to find the points of intersection and integrate the difference between the two curves over the interval.
Finding Points of Intersection:
To find the points of intersection, we set the equations of the parabola and the line equal to each other and solve for x:
x^2 - 2 = x - 1
Rearranging the equation, we get:
x^2 - x - 1 = 0
Using the quadratic formula, we find the values of x:
x = [-(-1) ± √((-1)^2 - 4(1)(-1))]/(2(1))
x = [1 ± √(1 + 4)]/2
x = [1 ± √5]/2
Since the line x = 0 intersects the parabola at (0, -2), we only need to consider the positive value of x:
x = (1 + √5)/2
Integrating the Difference:
To find the area bounded by the curves, we need to integrate the difference between the two functions over the interval [0, (1 + √5)/2]:
Area = ∫[(x - 1) - (x^2 - 2)]dx (from x = 0 to x = (1 + √5)/2)
= ∫(2 - x - x^2)dx (from x = 0 to x = (1 + √5)/2)
= [2x - (x^2)/2 - (x^3)/3] (from x = 0 to x = (1 + √5)/2)
Evaluating the integral at the upper and lower limits, we get:
Area = [2((1 + √5)/2) - ((1 + √5)/2)^2/2 - ((1 + √5)/2)^3/3] - [2(0) - (0^2)/2 - (0^3)/3]
= [(1 + √5) - ((1 + √5)/2)^2 - ((1 + √5)/2)^3/3]
Simplifying further, we get:
Area = 15/2 - (5 + 2√5)/4 - (1 + √5)/6
= 15/2 - (15 + 6√5 + 2 + √5)/12
= 15/2 - (17 + 6√5)/12
= (90 - 17 - 6√5)/12
= (73 - 6√5)/12
Therefore, the correct answer is option B, 15/2.
Finding Points of Intersection:
To find the points of intersection, we set the equations of the parabola and the line equal to each other and solve for x:
x^2 - 2 = x - 1
Rearranging the equation, we get:
x^2 - x - 1 = 0
Using the quadratic formula, we find the values of x:
x = [-(-1) ± √((-1)^2 - 4(1)(-1))]/(2(1))
x = [1 ± √(1 + 4)]/2
x = [1 ± √5]/2
Since the line x = 0 intersects the parabola at (0, -2), we only need to consider the positive value of x:
x = (1 + √5)/2
Integrating the Difference:
To find the area bounded by the curves, we need to integrate the difference between the two functions over the interval [0, (1 + √5)/2]:
Area = ∫[(x - 1) - (x^2 - 2)]dx (from x = 0 to x = (1 + √5)/2)
= ∫(2 - x - x^2)dx (from x = 0 to x = (1 + √5)/2)
= [2x - (x^2)/2 - (x^3)/3] (from x = 0 to x = (1 + √5)/2)
Evaluating the integral at the upper and lower limits, we get:
Area = [2((1 + √5)/2) - ((1 + √5)/2)^2/2 - ((1 + √5)/2)^3/3] - [2(0) - (0^2)/2 - (0^3)/3]
= [(1 + √5) - ((1 + √5)/2)^2 - ((1 + √5)/2)^3/3]
Simplifying further, we get:
Area = 15/2 - (5 + 2√5)/4 - (1 + √5)/6
= 15/2 - (15 + 6√5 + 2 + √5)/12
= 15/2 - (17 + 6√5)/12
= (90 - 17 - 6√5)/12
= (73 - 6√5)/12
Therefore, the correct answer is option B, 15/2.
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The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is :a)15/4b)15/2c)21/2d)17/4Correct answer is option 'B'. Can you explain this answer?
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