Question

If ${\int }_a^b{x}^{3}dx=0$ and ${\int }_a^b{x}^{2}dx=\frac{2}{3}$, then what are the values of $a$ and $b$ respectively?

• A. $-1,1$
• B. $1,1$
• C. $0,0$
• D. $2,-2$

Answer 1

The correct option is A $-1,1$

from given equation , we can get

$b^4-a^4=0\Rightarrow (b-a)(b+a)(b^2+a^2)=0$...............(1)

and $\frac{b^3-a^3}{3}=\frac{2}{3}\Rightarrow b^3-a^3=2\Rightarrow (b-a)(b^2+ab+a^2)=2$...............(2)

From (2) we can see that $(b-a)$ is not equal to 0 and $(b^2+a^2)$ cannot be zero as it is always positive in (1)

so, in (1) we can see that $(b+a)=0\Rightarrow a=-b$

Putting $a=-b$ in (2)

$2b\cdot b^2=2\Rightarrow b=1or-1$

so. $a=-1or1$

Therefore, Answer is $A$