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The area of the region bounded by the parabola y = x2 + 1 and the straight line x + y = 3 is given by
- a)45/7
- b)25/4
- c)π/18
- d)9/2
Correct answer is option 'D'. Can you explain this answer?
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The area of the region bounded by the parabola y = x2 + 1 and the stra...
Understanding the Problem:
The given problem involves finding the area of the region bounded by the parabola y = x^2 + 1 and the straight line x + y = 3. To find the area, we need to determine the points of intersection of the parabola and the line.
Finding Points of Intersection:
To find the points of intersection, we first need to substitute y = x^2 + 1 into x + y = 3.
This gives us x + x^2 + 1 = 3, which simplifies to x^2 + x - 2 = 0.
Factoring this quadratic equation, we get (x + 2)(x - 1) = 0.
Therefore, x = -2 or x = 1.
Substitute these values back into y = x^2 + 1 to find the corresponding y-coordinates.
Calculating Area:
To find the area of the region bounded by the parabola and the line, we need to integrate the difference in y-coordinates with respect to x from -2 to 1.
The area is given by the integral of (3 - x - x^2 - 1) dx from -2 to 1.
This simplifies to the integral of (-x^2 - x + 2) dx from -2 to 1.
Integrating this expression gives us [-x^3/3 - x^2/2 + 2x] evaluated from -2 to 1.
Evaluating at x = 1 and x = -2, we get the area as 9/2.
Therefore, the correct answer is option 'D' (9/2).
The given problem involves finding the area of the region bounded by the parabola y = x^2 + 1 and the straight line x + y = 3. To find the area, we need to determine the points of intersection of the parabola and the line.
Finding Points of Intersection:
To find the points of intersection, we first need to substitute y = x^2 + 1 into x + y = 3.
This gives us x + x^2 + 1 = 3, which simplifies to x^2 + x - 2 = 0.
Factoring this quadratic equation, we get (x + 2)(x - 1) = 0.
Therefore, x = -2 or x = 1.
Substitute these values back into y = x^2 + 1 to find the corresponding y-coordinates.
Calculating Area:
To find the area of the region bounded by the parabola and the line, we need to integrate the difference in y-coordinates with respect to x from -2 to 1.
The area is given by the integral of (3 - x - x^2 - 1) dx from -2 to 1.
This simplifies to the integral of (-x^2 - x + 2) dx from -2 to 1.
Integrating this expression gives us [-x^3/3 - x^2/2 + 2x] evaluated from -2 to 1.
Evaluating at x = 1 and x = -2, we get the area as 9/2.
Therefore, the correct answer is option 'D' (9/2).
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The area of the region bounded by the parabola y = x2 + 1 and the straight line x + y = 3 is given bya)45/7b)25/4c)π/18d)9/2Correct answer is option 'D'. Can you explain this answer?
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