The value of $\int_{-2}^{3}\left|1-x^{2}\right| d x$ is

JEE Main · 2004 · 24 Apr 2004 · Offline · Definite Integrals

The value of

\int_{- 2}^{3} 1 - x^{2} d x

Answer: (A)

The quantity

1 - x^{2}

changes sign at

x = \pm 1

, so split the range: it is negative on

[- 2, - 1]

, non-negative on

[- 1, 1]

, and negative again on

[1, 3]

. Thus

\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

. These evaluate to

\frac{4}{3}

\frac{4}{3}

and

\frac{20}{3}

, and their sum is

\frac{28}{3}

JEE Main · 2004 · 24 Apr 2004 · Offline · Single correct option

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