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Question

Mathematics Question on Area between Two Curves

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

A

23(2+1)\frac{2}{3}\left(\sqrt2+1 \right)

B

43(2−1)\frac{4}{3}\left(\sqrt2-1\right)

C

2(2−1)2\left(\sqrt2-1\right)

D

83(2−1)\frac{8}{3}\left(\sqrt2-1\right)

Answer

83(2−1)\frac{8}{3}\left(\sqrt2-1\right)

Explanation

Solution

The correct answer is (D) : 83(2−1)\frac{8}{3}\left(\sqrt2-1\right)

Fig.

Area =2∫02(1−∣x2−1∣) dx=2 \int_{0}^{\sqrt{2}} (1 - |x^2 - 1|) \,dx
=2[∫01(1−(1−x2)) dx+∫12(2−x2) dx]=2 \left[ \int_{0}^{1} (1 - (1 - x^2)) \,dx + \int_{1}^{\sqrt{2}} (2 - x^2) \,dx \right]
2[[x33]01+[2x−x33]12]2 \left[ \left[ \frac{x^3}{3} \right]_{0}^{1} + \left[ 2x - \frac{x^3}{3} \right]_{1}^{\sqrt{2}} \right]
=2(42−43)= 2\left(\frac{4\sqrt2-4}{3}\right)
=83(2−1)=\frac{8}{3}\left(\sqrt2-1\right)