Concept:

Integral properties: Consider a function f(x) defined on x.

• \(\mathop \smallint \limits_{ - {\rm{a}}}^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{\;dx}} = \left\{ {\begin{array}{*{20}{c}} {2\mathop \smallint \limits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}},\;\;\;f\left( {\rm{x}} \right) = f\left( { - x} \right)}\\ {0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;f\left( {\rm{x}} \right) = - f\left( { - x} \right)} \end{array}} \right.\)

Calculation:

Given:

\(\mathop \smallint \nolimits_{\rm{a}}^{\rm{b}} {{\rm{x}}^3}{\rm{dx}} = 0{\rm{\;}}\)

\(\mathop \smallint \nolimits_{\rm{a}}^{\rm{b}} {{\rm{x}}^2}{\rm{dx}} = \frac{2}{3}\)

\(\mathop \smallint \nolimits_{\rm{a}}^{\rm{b}} {{\rm{x}}^3}{\rm{dx}} = 0{\rm{\;}}\)

Let f(x) = x3

F(-x) = -x3

⇒ f(-x) = - f(x)

∴ f(x) is an odd function.

We know that, if f(x) is odd function then,\(\mathop \smallint \nolimits_{ - {\rm{a}}}^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}} = 0\)

So, b = -a ----(1)

Now,\(\mathop \smallint \nolimits_{\rm{a}}^{\rm{b}} {{\rm{x}}^2}{\rm{dx}} = \frac{2}{3}\)

\(\Rightarrow \left[ {\frac{{{{\rm{x}}^3}}}{3}} \right]_{\rm{a}}^{\rm{b}} = \frac{2}{3}\)

⇒ b3 – a3 = 2

⇒ -a3 – a3 = 2 [Using (1)]

⇒ a3 = -1

⇒ a = -1

So, b = 1