The values of the integrals $$\int\limits_0^1 {\left( {\int\limits_0^1 {{{x - y} \over {{{\left( {x + y} \right)}^3}}}dy} } \right)} dx\,\,$$ and $$\,\,\int\limits_0^1 {\left( {\int\limits_0^1 {{{x - y} \over {{{\left( {x + y} \right)}^3}}}dx} } \right)} dy\,\,$$ are

Let $$\,\,f\left( x \right) = {e^{x + {x^2}}}\,\,$$ for real $$x.$$ From among the following. Choose the Taylor series approximation of $$f$$ $$(x)$$ around $$x=0,$$ which includes all powers of $$x$$ less than or equal to $$3.$$

A three dimensional region $$R$$ of finite volume is described by $$\,\,{x^2} + {y^2} \le {z^3},\,\,\,0 \le z \le 1$$
Where $$x, y, z$$ are real. The volume of $$R$$ correct to two decimal places is __________.

The integral $$\,\,{1 \over {2\pi }}\int {\int_D {\left( {x + y + 10} \right)dxdy\,\,} } $$ where $$D$$ denotes the disc: $${x^2} + {y^2} \le 4,$$ evaluates to _________.