Area of the region bounded by the curve y=x|x | vedclass

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Question

(B) The curve is given by $y = x|x|$.Since the interval is $x \in [0, 1]$,we have $x \ge 0$,so $|x| = x$.Thus,the curve becomes $y = x \cdot x = x^2$

Vedclass · Accepted Answer

$\frac{1}{3}$

Vedclass · Answer

(B) The curve is given by $y = x|x|$.Since the interval is $x \in [0, 1]$,we have $x \ge 0$,so $|x| = x$.Thus,the curve becomes $y = x \cdot x = x^2$ for $x \in [0, 1]$.The area $A$ bounded by the curve,the $X$-axis,and the lines $x=0$ and $x=1$ is given by the definite integral:$A = \int_{0}^{1} |y| \, dx = \int_{0}^{1} x^2 \, dx$.Evaluating the integral:$A = \left[ \frac{x^3}{3} \right]_{0}^{1} = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} - 0 = \frac{1}{3}$.Therefore,the area is $\frac{1}{3}$ sq. units.

Area of the region bounded by the curve $y=x|x|$,$X$-axis and lines $x=0$ and $x=1$ is . . . . . . sq. units.

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