Question

${\int }_a^b{x}^{3}dx=0$ and ${\int }_a^b{x}^{2}dx=\frac{2}{3}$: Find both ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{b}^{\prime }$.

Answer 1

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

$\therefore {|}_a^b\frac{x^4}{4}=0$

$\therefore b^4-a^4=0\times 4$

$\therefore b^4=a^4$

$\therefore b=\pm a$

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

$\therefore \frac{x^3}{3}{|}_a^b=\frac{2}{3}$

$\therefore b^3-a^3=2$

Now if $b=a$

$b^3-a^3=0\to$ Not possible

Now if $b=-a$

$\therefore -a^3-a^3=2$

$\therefore -2a^3=2$

$\therefore a=-1$

$\therefore b=1$

$\therefore a=-1b=1$