The area bounded by the curves y=|x^2-1| and y=1 is:

Question

The area bounded by the curves $y=|x^2-1|$ and $y=1$ is:

Accepted Answer

Area $$ = 2\int\limits_0^{\sqrt 2 } {(1 - |{x^2} - 1|)dx} $$

$$2\left[ {\int\limits_0^1 {\left( {1 - (1 - {x^2})} \right)dx + \int\limits_1^{\sqrt 2 } {\left( {2 - {x^2}} \right)dx} } } \right]$$

$$ = 2\left[ {\left. {\left[ {{{{x^3}} \over 3}} \right]_0^1 + \left[ {2x - {{{x^3}} \over 3}} \right]_1^{\sqrt 2 }} \right]} \right]$$

$$ = 2\left( {{{4\sqrt 2  - 4} \over 3}} \right) = {8 \over 3}\left( {\sqrt 2  - 1} \right)$$

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