Question

If ${\int }_a^b{x}^{3}dx=0\text{and}if{\int }_a^b{x}^{2}dx=\frac{2}{3}$, find $a$ and $b$.

Answer 1

${\int }_a^b{x}^{3}dx=0$

${\left[\frac{x^4}{4}\right]}_a^b=\frac{b^4-a^4}{4}=0$

$\Rightarrow b^4-a^4=0$

$\Rightarrow b=\pm a$

${\int }_a^b{x}^{2}dx=\frac{2}{3}$

${\left[\frac{x^3}{3}\right]}_a^b=\frac{2}{3}$

$\frac{b^3-a^3}{3}=\frac{2}{3}$

$\Rightarrow {\left(a\right)}^3-a^3\ne 2$ or ${(-a)}^3-a^3=2$ since $b=\pm a$

$\Rightarrow {-a}^3-a^3=2$

$\Rightarrow -2a^3=2$

$\Rightarrow a^3=-1$

$\therefore a=-1$

$\Rightarrow b=\pm a=\pm 1$

Hence $a=-1,b=\pm 1$