Let $f(x)$ be a continuous function such that $\int\limits_n^{n+1} f(x) d x=n^3, n \in Z$. Then, the value of the integral $\int\limits_{-3}^3 f(x) d x$, is
Answer & explanation
Correct answer: option 2
We have,
$\int\limits_{-3}^3 f(x) d x=\sum\limits_{r=0}^5 \int\limits_{-3+r}^{-3+r+1} f(x) d x$
$\Rightarrow \int\limits_{-3}^3 f(x) d x=\sum\limits_{r=0}^5(-3+r)^3$ $\left[∵ \int\limits_n^{n+1} f(x) d x=n^3\right]$
$\Rightarrow \int\limits_{-3}^3 f(x) d x=\left[(-3)^3+(-2)^3+(-1)^3+0^2+1^3+2^3\right]=-27$