Question

The area bounded by the curves y = |x² – 1| and y = 1 is

• A. \(\frac{2}{3}\left(\sqrt2+1 \right)\)
• B. \(\frac{4}{3}\left(\sqrt2-1\right)\)
• C. \(2\left(\sqrt2-1\right)\)
• D. \(\frac{8}{3}\left(\sqrt2-1\right)\)

Answer 1

The Correct Option is D

The correct answer is (D) : \(\frac{8}{3}\left(\sqrt2-1\right)\)

Fig.

Area \(=2 \int_{0}^{\sqrt{2}} (1 - |x^2 - 1|) \,dx\)
\(=2 \left[ \int_{0}^{1} (1 - (1 - x^2)) \,dx + \int_{1}^{\sqrt{2}} (2 - x^2) \,dx \right]\)
\(2 \left[ \left[ \frac{x^3}{3} \right]_{0}^{1} + \left[ 2x - \frac{x^3}{3} \right]_{1}^{\sqrt{2}} \right]\)
\(= 2\left(\frac{4\sqrt2-4}{3}\right)\)
\(=\frac{8}{3}\left(\sqrt2-1\right)\)