The area of the region bounded by the curve y | vedclass

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(C) The curve is given by $y = x|x|$.We can define this piecewise as:$y = \begin{cases} x^2, & 	ext{if } x \ge 0 \ -x^2, & 	ext{if } x < 0 \end{cas

Vedclass · Accepted Answer

$2/3$

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(C) The curve is given by $y = x|x|$.We can define this piecewise as:$y = \begin{cases} x^2, & 	ext{if } x \ge 0 \ -x^2, & 	ext{if } x The area $A$ is given by the integral $\int_{-1}^{1} |y| \, dx$.Since the function $y = x|x|$ is an odd function,the area bounded by the curve and the $x$-axis from $x=-1$ to $x=1$ is calculated as:$A = \int_{-1}^{0} | -x^2 | \, dx + \int_{0}^{1} | x^2 | \, dx$$A = \int_{-1}^{0} x^2 \, dx + \int_{0}^{1} x^2 \, dx$$A = [\frac{x^3}{3}]_{-1}^{0} + [\frac{x^3}{3}]_{0}^{1}$$A = (0 - (-\frac{1}{3})) + (\frac{1}{3} - 0)$$A = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}$ square units.

The area of the region bounded by the curve $y=x|x|$,lines $x=-1$ and $x=1$ is . . . . . . .

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