Let $$f\left( x \right)$$  be a continuous function such that $$\int_n^{n + 1} {f\left( x \right)dx = {n^3},\,n\, \in \,Z.} $$      Then the value of $$\int_{ - 3}^3 {f\left( x \right)dx} $$    is :

Solution :
$$\eqalign{ & {\text{Given,}} \cr & \int_n^{n + 1} {f\left( x \right)} dx = {n^3}\,;\,n \in {\bf{Z}}......\left( 1 \right) \cr & {\text{Let }}I = \int_{ - 3}^3 {f\left( x \right)dx} \cr & = \int_{ - 3}^{ - 2} {f\left( x \right)} dx + \int_{ - 2}^{ - 1} {f\left( x \right)dx + } \int_{ - 1}^0 {f\left( x \right)dx} + \int_0^1 {f\left( x \right)dx} + \int_1^2 {f\left( x \right)dx} + \int_2^3 {f\left( x \right)dx} \cr & = {\left( { - 3} \right)^3} + {\left( { - 2} \right)^3} + {\left( { - 1} \right)^3} + {0^3} + {1^3} + {2^3} \cr & = - 27 - 8 - 1 + 0 + 1 + 8 \cr & = - 27 \cr & {\text{Hence, }}\int_{ - 3}^3 {f\left( x \right)dx} = - 27 \cr} $$