Find the area between the curves y=x and y=x^ | vedclass

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Question

(A) The required area is bounded by the curves $y=x$ and $y=x^{2}$.First,find the points of intersection by setting $x = x^{2}$,which gives $x^{2} - x

Vedclass · Accepted Answer

$\frac{1}{6}$

Vedclass · Answer

(A) The required area is bounded by the curves $y=x$ and $y=x^{2}$.First,find the points of intersection by setting $x = x^{2}$,which gives $x^{2} - x = 0$,so $x(x-1) = 0$. Thus,the intersection points are $x=0$ and $x=1$.In the interval $[0, 1]$,the line $y=x$ lies above the parabola $y=x^{2}$.The area $A$ is given by the integral:$A = \int_{0}^{1} (x - x^{2}) dx$Evaluating the integral:$A = \left[ \frac{x^{2}}{2} - \frac{x^{3}}{3} \right]_{0}^{1}$$A = \left( \frac{1}{2} - \frac{1}{3} \right) - (0 - 0)$$A = \frac{3-2}{6} = \frac{1}{6}$ sq. units.

Find the area between the curves $y=x$ and $y=x^{2}$.

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