Question

The value of $ \int_{-2}^{3}{|1-{{x}^{2}}|}\,dx $ is

• A. $ \frac{7}{3} $
• B. $ \frac{14}{3} $
• C. $ \frac{28}{3} $
• D. $ \frac{1}{3} $

Answer 1

Answer: (C)

$ \int_{-2}^{3}{|1-{{x}^{2}}|}dx $ $ =\int_{-2}^{-1}{({{x}^{2}}-1)}dx+\int_{-1}^{1}{(1-{{x}^{2}})}dx $ $ +\int_{1}^{3}{({{x}^{2}}-1)}dx $ $ =\left[ \frac{{{x}^{3}}}{3}-x \right]_{-2}^{-1}+\left[ x-\frac{{{x}^{3}}}{3} \right]_{-1}^{1}+\left[ \frac{{{x}^{3}}}{3}-x \right]_{1}^{3} $ $ =\frac{2}{3}+\frac{2}{3}+2\left( \frac{2}{3} \right)+(9-3)-\left( \frac{1}{3}-1 \right) $ $ =\frac{28}{3} $