91Ó°ÊÓ

Nested integration, often referred to as iterated integration, is a method where we perform integrals over more than one variable. This process involves repeatedly integrating a function, taking into account different variables in a specific order. In our problem, we have been given an iterated integral, \[ \int_{1}^{4} \int_{0}^{4} \left( \frac{x}{2}+\sqrt{y} \right) dx \, dy \]. Here, the integration is first conducted with respect to the variable \( x \), while \( y \) is held constant, followed by integration with respect to \( y \).

• The procedure simplifies a complex integral into easier, sequential steps.
• It highlights the structure of the problem as each step tackles one layer of the integral.

This division allows a more systematic way of handling multidimensional integrals, which are common in higher-level calculus and physics.

The inner integral is the first step in a nested integration process. It refers to integrating the function with respect to one variable while considering others as constants. In this problem, the inner integral is \[ \int_{0}^{4} \left( \frac{x}{2} + \sqrt{y} \right) dx \].
Firstly, separate the terms in the integral:

• \( \int_{0}^{4} \frac{x}{2} \, dx \)
• \( \int_{0}^{4} \sqrt{y} \, dx \)

The integral \( \int_{0}^{4} \frac{x}{2} \, dx \) is solved by the power rule, resulting in \[ \frac{x^2}{4} \Big|_0^4 = 4 \]. For the second integral, since \( \sqrt{y} \) is treated as a constant relative to \( x \), the integral becomes simple multiplication yielding \[ 4\sqrt{y} \]. The overall evaluation of the inner integral thus sums to \[ 4 + 4\sqrt{y} \].

The outer integral is evaluated after computing the inner integral and involves integrating over the remaining variable. Once the inner integral gives us the function \( 4 + 4\sqrt{y} \), our task is to integrate it from \( y=1 \) to \( y=4 \). The outer integral becomes \[ \int_{1}^{4} (4 + 4\sqrt{y}) \, dy \].
Here, separate the expression into two simpler integrals:

• \( \int_{1}^{4} 4 \, dy \)
• \( \int_{1}^{4} 4\sqrt{y} \, dy \)

The first part, \( \int_{1}^{4} 4 \, dy \), straightforwardly results in \[ 4(4 - 1) = 12 \]. For the second part, use the power rule to find \[ 4\int_{1}^{4} y^{1/2} \, dy = 4 \left[ \frac{2}{3} y^{3/2} \right]_1^4 = \frac{56}{3} \]. Adding these results, compute the final result of the integration.

Evaluating definite integrals involves finding the accumulation of the function's values over a specified interval, resulting in a numerical value. In our problem, after performing the necessary integration processes, we add the results from the outer integral.

• The first part of the outer integral gives us \( 12 \).
• The second evaluation yields \( \frac{56}{3} \).

To combine these, express \( 12 \) as a fraction: \( \frac{36}{3} \) so that the fractions can be added easily. The sum then is \[ \frac{36}{3} + \frac{56}{3} = \frac{92}{3} \]. Thus, the value of the entire iterated integral is \[ \frac{92}{3} \]. This represents the total accumulated value over the defined region, emphasizing how nested and definite integrals can help solve complex multifaceted problems.