The area bounded by the parabola y=x^2 and th | vedclass

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(D) To find the area bounded by the parabola $y=x^2$ and the line $y=x$,we first determine their points of intersection by setting $x^2 = x$.This give

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$\frac{1}{6}$ sq. units

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(D) To find the area bounded by the parabola $y=x^2$ and the line $y=x$,we first determine their points of intersection by setting $x^2 = x$.This gives $x^2 - x = 0$,which implies $x(x-1) = 0$.Thus,the points of intersection are $x=0$ and $x=1$.For $x \in [0, 1]$,the line $y=x$ lies above the parabola $y=x^2$.The required area $A$ is given by the integral:$A = \int_0^1 (x - x^2) dx$$A = \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_0^1$$A = \left( \frac{1^2}{2} - \frac{1^3}{3} \right) - (0 - 0)$$A = \frac{1}{2} - \frac{1}{3} = \frac{3-2}{6} = \frac{1}{6}$ sq. units.

The area bounded by the parabola $y=x^2$ and the line $y=x$ is

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