Q.Find the area of the regions bounded by the line $y = -2x$, the X-axis and the lines $x = -1$ and $x = 2$.

Split at $x=0$ where $y=-2x$ crosses the axis; take each piece's area as positive: $1 + 4 = 5$ square units.

The line is $y = -2x$. It meets the X-axis at $x=0$. Between the given limits it changes sign:

• on $[-1,\,0]$, $y = -2x \ge 0$ (region above the axis),
• on $[0,\,2]$, $y = -2x \le 0$ (region below the axis).

Because area is always positive, we integrate the absolute value $|y|$, splitting the interval at $x=0$.

Part 1 — on $[-1,0]$ (curve above axis):

$A_1 = \int_{-1}^{0} (-2x)\,dx = \left[-x^2\right]_{-1}^{0} = 0 - \big(-(-1)^2\big) = 0 - (-1) = 1.$

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