First slide

Evaluation of definite integrals

Question

$\int_{- 2}^{3} | 1 - x^{2} | d x =$

Difficult

A

$\frac{28}{3}$
B

$\frac{14}{3}$
C

$\frac{7}{3}$
D

$\frac{1}{3}$

Solution

$\int_{- 2}^{3} | 1 - x^{2} | d x = \int_{- 2}^{- 1} (x^{2} - 1) d x + \int_{- 1}^{1} (1 - x^{2}) d x + \int_{1}^{3} (x^{2} - 1) d x$
${= \frac{x^{3}}{3} |}_{- 2}^{- 1} - 1 + 2 - {\frac{x^{3}}{3} |}_{- 1}^{1} + {\frac{x^{3}}{3} |}_{1}^{3} - 2$

$= \frac{7}{3} - \frac{5}{3} + \frac{26}{3} = \frac{28}{3}$

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