If \( \int_{a}^{b}\displaystyle x^3dx=0 \) and \( \int_{a}^{b}\displaystyle x^2dx=\displaystyle\frac{2}{3} \), then the values of \( a \) and \( b \) are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( a=-1 \) and \( b=1 \)
Both integrals are simple powers of x:
\( \displaystyle \int_{a}^{b}x^3\,dx=\frac{b^4-a^4}{4}=0 \)
\( \displaystyle b^4=a^4\ \Rightarrow\ b=\pm a \)
\( \displaystyle \int_{a}^{b}x^2\,dx=\frac{b^3-a^3}{3}=\frac{2}{3} \)
\( \displaystyle b^3-a^3=2 \)
If b = a, then \( b^3-a^3=0\ne 2 \). So b = −a:
\( \displaystyle -a^3-a^3=2 \)
\( \displaystyle a^3=-1\ \Rightarrow\ a=-1 \)
\( \displaystyle b=1 \)
Shortcut: Substitute the options in the second integral: equal limits (Options (3), (4)) give 0, and reversing the limits (Option (2)) gives
\( \displaystyle \int_{1}^{-1}x^2\,dx=-\frac{2}{3} \)
Only Option (1) gives \( \frac{2}{3} \), and \( \int_{-1}^{1}x^3\,dx=0 \) as x3 is odd.