Question

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

Hint:

If \( \int_a^b f(x) dx = 0 \), check for symmetry of \( f(x) \). Odd functions integrate to zero over symmetric limits.

Answer 1

Step 1: Evaluate the given integral. \[ \int_a^b x^3 dx = \left[ \frac{x^4}{4} \right]_a^b = 0. \] \[ \frac{b^4}{4} - \frac{a^4}{4} = 0. \] \[ b^4 = a^4. \] \[ b = -a \quad \text{(since \( b \neq a \))}. \] Step 2: Solve for \( a \) and \( b \). \[ \int_a^b x^2 dx = \left[ \frac{x^3}{3} \right]_a^b = \frac{2}{3}. \] Substituting \( b = -a \): \[ \frac{(-a)^3}{3} - \frac{a^3}{3} = \frac{2}{3}. \] \[ \frac{-a^3}{3} - \frac{a^3}{3} = \frac{2}{3}. \] \[ \frac{-2a^3}{3} = \frac{2}{3}. \] \[ -2a^3 = 2. \] \[ a^3 = -1. \] \[ a = -1, \quad b = 1. \] Final Answer: \( a = -1, b = 1 \).