Question

If \(\int_a^b x^3 \, dx = 0\) and \(\int_a^b x^2 \, dx = \frac{2}{3}\), then the values of \(a\) and \(b\) respectively are

• A. $1$ and $1$
• B. $0$ and $1$
• C. $1$ and $-1$
• D. $-1$ and $0$
• E. $-1$ and $1$

Hint:

If $\int_a^b$ of an odd function is $0$, then limits are symmetric: - $a = -k$, $b = k$ Use this shortcut to save time in exams.

Answer 1

Answer: (E)

Concept: Use definite integration formulas: \[ \int_a^b x^n dx = \frac{b^{n+1} - a^{n+1}}{n+1} \] Also: - $x^3$ is an odd function - $x^2$ is an even function

Step 1: Evaluate $\int_a^b x^3 dx = 0$.
\[ \int_a^b x^3 dx = \frac{b^4 - a^4}{4} = 0 \] \[ \Rightarrow b^4 = a^4 \Rightarrow b = -a \]

Step 2: Use second condition $\int_a^b x^2 dx = \frac{2}{3}$.
\[ \int_a^b x^2 dx = \frac{b^3 - a^3}{3} \] Substitute $b = -a$: \[ = \frac{(-a)^3 - a^3}{3} = \frac{-a^3 - a^3}{3} = \frac{-2a^3}{3} \]

Step 3: Equate with given value.
\[ \frac{-2a^3}{3} = \frac{2}{3} \] \[ \Rightarrow -2a^3 = 2 \Rightarrow a^3 = -1 \Rightarrow a = -1 \]

Step 4: Find $b$.
\[ b = -a = 1 \]