Chapter: Application of Definite Integration· Mathematics & Statistics (Commerce) Part 1

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

Q.2 (B)(iii) Find the area of the regions bounded by the line \(y = -2x\), the X-axis and the lines \(x = -1\) and \(x = 2\).

Doubtly Asked 29 Oct 2026 Board exam · March 2024 hsc 12th Save WhatsApp Report

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Answer: 5 sq. units

[Rough sketch: the line \(y = -2x\) passes through the origin; for \(-1 \le x \le 0\) it lies above the X-axis and for \(0 \le x \le 2\) it lies below the X-axis.]

Area above the X-axis: \(A_1 = \displaystyle\int_{-1}^{0} (-2x)\,dx = \left[-x^2\right]_{-1}^{0} = 0 - (-1) = 1\).

Area below the X-axis: \(\displaystyle\int_{0}^{2} (-2x)\,dx = \left[-x^2\right]_{0}^{2} = -4\), so \(A_2 = |-4| = 4\).

Required area \(= A_1 + A_2 = 1 + 4 = 5\) sq. units.

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

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