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Theorems on Integration

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Question

Solve: $\int_{- 1}^{1} | 1 - x^{2} | d x$

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Solution

Now,
$\int_{- 1}^{1} | 1 - x^{2} | d x$
$= \int_{- 1}^{1} (1 - x^{2}) d x$ [ Since $1 - x^{2} \geq 0$ gives $x^{2} \leq 1$ gives $- 1 \leq x \leq 1$ so in $[- 1, 1]$ we have $| 1 - x^{2} | = 1 - x^{2}$ ]
$= {[x - \frac{x^{3}}{3}]}_{- 1}^{1}$
$= 2 - \frac{2}{3}$
$= \frac{4}{3}$

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