The area of the region bounded by the parabol | vedclass

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Question

(B) The given curves are $y=x^2$ and $y=|x|$.Since both curves are symmetric about the $y$-axis,the total area is twice the area in the first quadrant

Vedclass · Accepted Answer

$\frac{1}{3}$ sq. units

Vedclass · Answer

(B) The given curves are $y=x^2$ and $y=|x|$.Since both curves are symmetric about the $y$-axis,the total area is twice the area in the first quadrant.In the first quadrant,$y=|x|$ becomes $y=x$.The intersection points are found by setting $x^2 = x$,which gives $x(x-1)=0$,so $x=0$ and $x=1$.The required area is given by:$	ext{Area} = 2 \int_0^1 (x - x^2) dx$$= 2 \left[ \frac{x^2}{2} - \frac{x^3}{3} ight]_0^1$$= 2 \left( \frac{1}{2} - \frac{1}{3} ight)$$= 2 \left( \frac{3-2}{6} ight) = 2 \left( \frac{1}{6} ight) = \frac{1}{3} 	ext{ sq. units}$.

The area of the region bounded by the parabola $y=x^2$ and the curve $y=|x|$ is

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