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When tackling iterated integrals, integration is a crucial tool that allows us to handle multi-variable functions. Iterated integrals involve integrating a function over a two-dimensional region or more. This means we perform integration multiple times, each time concerning a different variable. Typically, for double integrals, this involves integrating first with respect to one variable while treating the other variable(s) as constants, followed by integrating with respect to the next variable.

In our given problem, \[\int_{0}^{1} \int_{2x}^{3x} (x+y^2) \, dy \, dx\], the process of integration starts with the inner integral \[\int_{2x}^{3x} (x+y^2) \, dy\]. Here, integration is performed with respect to the variable \( y \) first, while treating \( x \) as a constant. This is followed by integrating the result with respect to \( x \) over the given limits from 0 to 1.
Integration here is about finding the sum of an infinite number of infinitesimally small areas, which contributes to the final accumulated value over the specified range. Essentially, integration allows us to determine the total accumulation, which, in the case of iterated integrals, provides the area under the surface defined by the function over the specified domain.

To successfully integrate a function, finding the antiderivative is a key step. An antiderivative of a function \( f(x) \) is another function \( F(x) \) such that the derivative of \( F(x) \) is \( f(x) \). In integrals, especially definite integrals, we use antiderivatives to calculate the area under a curve over a specified range.

For instance, the inner integral, \[\int_{2x}^{3x} (x+y^2) \, dy\] requires finding the antiderivatives of both \( x \) and \( y^2 \) with respect to \( y \). Here, the antiderivative of \( x \) with respect to \( y \) is \( xy \), as \( x \) is treated as a constant. On the other hand, the antiderivative of \( y^2 \) with respect to \( y \) is \( \frac{y^3}{3} \). Finding these antiderivatives allows us to evaluate the function at the given limits of integration and proceed with the problem.
Using antiderivatives simplifies the process of integration by converting a differentiation problem into one involving evaluation and subtraction between bounds. This transition is vital because it translates complex integrals into manageable computations.

Limits of integration define the boundaries over which the integration is performed. In iterated integrals, it's critical to correctly understand these limits for each variable, as they determine the region of integration.

For the given problem, the iterated integral\[\int_{0}^{1} \int_{2x}^{3x} (x+y^2) \, dy \, dx\]involves limits associated with both variables, \( x \) and \( y \). The inner integral, which involves \( y \), has limits from \( 2x \) to \( 3x \). These limits indicate that for each value of \( x \), \( y \) starts at \( 2x \) and ends at \( 3x \).
The outer integral, concerning \( x \), has limits from 0 to 1, suggesting that \( x \) varies between these values across the entire domain specified by the problem. To solve such integrals, it's essential to substitute these boundary values correctly into the antiderivatives to obtain the definite integral's result.
Incorrectly managing the limits during integration can lead to erroneous results, as they precisely determine the real geometric region over which integration occurs. Understanding and applying these limits correctly ensures the accurate computation of the iterated integral.