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Question Number 161229 by cortano last updated on 14/Dec/21

$$Givenf\left(x\right)=\{\begin{array}{l}1-\mid x\mid ;x⩽1\\ \mid x\mid -1;x>1\end{array}$$$$find{\int }_{-3}^8[f(x-1)+f(x+1)]dx.$$

Answered by mr W last updated on 14/Dec/21

Commented by mr W last updated on 14/Dec/21

$$red:f\left(x\right)$$$$blue:f(x-1)$$$$green:f(x+1)$$$${\int }_{-3}^{8}f(x-1)dx=-\frac{3\times 3}{2}+\frac{2\times 1}{2}+\frac{6\times 6}{2}=\frac{29}{2}$$$${\int }_{-3}^{8}f(x+1)dx=-\frac{1\times 1}{2}+\frac{2\times 1}{2}+\frac{8\times 8}{2}=\frac{65}{2}$$$${\int }_{-3}^8[f(x-1)+f(x+1)]dx=\frac{29+65}{2}=47$$

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