If int_a^b x^3 dx = 0 and int_a^b x^2 dx = fr | vedclass

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(A) Given,$\int_a^b x^3 dx = 0$ and $\int_a^b x^2 dx = \frac{2}{3}$.First,evaluate the integral $\int_a^b x^3 dx = 0$:$\left[\frac{x^4}{4}\right]_a^b

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$a = -1$ and $b = 1$

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(A) Given,$\int_a^b x^3 dx = 0$ and $\int_a^b x^2 dx = \frac{2}{3}$.First,evaluate the integral $\int_a^b x^3 dx = 0$:$\left[\frac{x^4}{4}ight]_a^b = 0 \Rightarrow \frac{b^4 - a^4}{4} = 0 \Rightarrow b^4 = a^4$.This implies $b = a$ or $b = -a$. Since $a$ and $b$ are limits of integration,we assume $a 
eq b$,so $b = -a$.Next,evaluate the integral $\int_a^b x^2 dx = \frac{2}{3}$:$\left[\frac{x^3}{3}ight]_a^b = \frac{2}{3} \Rightarrow \frac{b^3 - a^3}{3} = \frac{2}{3} \Rightarrow b^3 - a^3 = 2$.Substitute $b = -a$ into the equation:$(-a)^3 - a^3 = 2 \Rightarrow -a^3 - a^3 = 2 \Rightarrow -2a^3 = 2$.$a^3 = -1 \Rightarrow a = -1$.Since $b = -a$,we have $b = -(-1) = 1$.Thus,$a = -1$ and $b = 1$.

If $\int_a^b x^3 dx = 0$ and $\int_a^b x^2 dx = \frac{2}{3}$,then

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