The area bounded by the line y=x,the x-axis,a | vedclass

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(B) The area bounded by the curve $y=x$,the $x$-axis,and the lines $x=-1$ and $x=2$ is given by the integral of the absolute value of $y$ with respect

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$ \frac{5}{2} $

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(B) The area bounded by the curve $y=x$,the $x$-axis,and the lines $x=-1$ and $x=2$ is given by the integral of the absolute value of $y$ with respect to $x$: $Area = \int_{-1}^{2} |y| \, dx = \int_{-1}^{2} |x| \, dx$ We split the integral at $x=0$ because the function changes sign: $Area = \int_{-1}^{0} |x| \, dx + \int_{0}^{2} |x| \, dx$ Since $|x| = -x$ for $x $Area = \int_{-1}^{0} (-x) \, dx + \int_{0}^{2} (x) \, dx$ $Area = \left[ -\frac{x^2}{2} ight]_{-1}^{0} + \left[ \frac{x^2}{2} ight]_{0}^{2}$ $Area = (0 - (-\frac{(-1)^2}{2})) + (\frac{2^2}{2} - 0)$ $Area = \frac{1}{2} + \frac{4}{2} = \frac{1}{2} + 2 = \frac{5}{2} 	ext{ sq. units.}$

The area bounded by the line $y=x$,the $x$-axis,and the ordinates $x=-1$ and $x=2$ is

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