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 the points of intersection by setting the equations equal t

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

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(D) To find the area bounded by the parabola $y=x^2$ and the line $y=x$,we first determine the points of intersection by setting the equations equal to each other:$x^2 = x$$x^2 - x = 0$$x(x - 1) = 0$This gives $x = 0$ and $x = 1$. The points of intersection are $O(0, 0)$ and $P(1, 1)$.In the interval $[0, 1]$,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$$A = \left[ \frac{x^2}{2} - \frac{x^3}{3} ight]_0^1$$A = \left( \frac{1^2}{2} - \frac{1^3}{3} ight) - (0 - 0)$$A = \frac{1}{2} - \frac{1}{3} = \frac{3 - 2}{6} = \frac{1}{6} 	ext{ sq. units}$

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

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