Question

Let $f\left(x\right)$ be a continuous function such that ${\int }_n^{n+1}f\left(x\right)dx=n^3,nϵZ.$

Then the value of ${\int }_{-3}^{3}f\left(x\right)dx$ is

• A. $9$
• B. $-27$
• C. $-9$
• D. none of these

Answer 1

Answer: (B)

$-27$

As ${\int }_n^{n}f\left(x\right)dx=n^3$

$\therefore {\int }_{-3}^{3}f\left(x\right)dx={\int }_{-3}^{-2}f\left(x\right)dx+{\int }_{-2}^{-1}f\left(x\right)dx+{\int }_{-1}^{0}f\left(x\right)dx$

$+{\int }_0^{1}f\left(x\right)dx+{\int }_1^{2}f\left(x\right)dx+{\int }_2^{3}f\left(x\right)dx={\left(3\right)}^3+{(-2)}^3+{(-1)}^3+{\left(0\right)}^3+{\left(1\right)}^2+{\left(2\right)}^3=-27$