The area enclosed between the parabala y = x2and the straight line y = x isa)1/8b)1/6c)1/3d)1/2Correct answer is option 'B'. Can you explain this answer?

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Answer

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Problem:
Find the area enclosed between the parabola y = x^2 and the straight line y = x.

Solution:
To find the area enclosed between the given parabola and the straight line, we need to determine the points of intersection first.

Finding the Points of Intersection:
To find the points of intersection, we need to equate the two equations:
x^2 = x

Rearranging the equation, we have:
x^2 - x = 0

Factoring out x, we get:
x(x - 1) = 0

So, x = 0 or x - 1 = 0
Therefore, x = 0 or x = 1

Hence, the points of intersection are (0, 0) and (1, 1).

Finding the Area:
To find the area enclosed between the parabola and the straight line, we integrate the difference of the two functions with respect to x over the interval [0, 1].

Area = ∫(x - x^2) dx, with limits 0 to 1

Integrating the function:
= [(x^2)/2 - (x^3)/3] evaluated from 0 to 1

Substituting the limits:
= [(1^2)/2 - (1^3)/3] - [(0^2)/2 - (0^3)/3]
= [1/2 - 1/3] - [0 - 0]
= 1/2 - 1/3
= (3 - 2)/6
= 1/6

Therefore, the area enclosed between the parabola y = x^2 and the straight line y = x is 1/6.

Hence, the correct answer is option 'B' - 1/6.

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