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Question Number 54028 by qw last updated on 28/Jan/19

$$\mathrm{If}\mathrm{for}\mathrm{every}\mathrm{integer}n,\underset{n}{\stackrel{n+1}{\int }}f\left(x\right)dx=n^2,$$$$\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\underset{-2}{\stackrel{4}{\int }}f\left(x\right)dx=$$

Answered by tanmay.chaudhury50@gmail.com last updated on 28/Jan/19

$${\int }_{-2}^{4}f\left(x\right)dx$$$${\int }_{-2}^{-2+1}f\left(x\right)dx+{\int }_{-1}^{-1+1}f\left(x\right)dx+{\int }_0^{0+1}f\left(x\right)dx+{\int }_1^{1+1}f\left(x\right)dx+{\int }_2^{2+1}f\left(x\right)dx+{\int }_3^{3+1}f\left(x\right)dx$$$$[{(-2)}^2+{(-1)}^2+{\left(0\right)}^2+{\left(1\right)}^2+{\left(2\right)}^2+{\left(3\right)}^2]$$$$=4+1+0+1+4+9$$$$=19$$

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