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(D) 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 the functi

Vedclass · Accepted Answer

$5/2$

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(D) 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 the function.$	ext{Area} = \int_{-1}^{2} |y| \, dx = \int_{-1}^{2} |x| \, dx$Since $|x| = -x$ for $x $	ext{Area} = \int_{-1}^{0} (-x) \, dx + \int_{0}^{2} (x) \, dx$Evaluating the integrals:$\int_{-1}^{0} (-x) \, dx = \left[ -\frac{x^2}{2} ight]_{-1}^{0} = 0 - \left( -\frac{(-1)^2}{2} ight) = 0 - (-1/2) = 1/2$$\int_{0}^{2} (x) \, dx = \left[ \frac{x^2}{2} ight]_{0}^{2} = \frac{2^2}{2} - 0 = 4/2 = 2$Total Area $= 1/2 + 2 = 5/2$.

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

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