Us to find the value of b given that f(x) = x³ and the value resulting from the Mean Value Theorem for Integrals for integral_a^b f(x) \,dx is 37.

The problem asks us to find the value of $b$ given that $f(x) = x^3$ and the value resulting from the Mean Value Theorem for Integrals for $\int_{a}^{b} f(x) \,dx$ is 37.

The Mean Value Theorem for Integrals states that for a continuous function $f$ on an interval $[a, b]$, there exists a number $c$ in $[a, b]$ such that:

$$\int_{a}^{b} f(x) \,dx = f(c)(b-a)$$

The average value of the function $f$ over the interval $[a, b]$ is given by:

$$$$