91Ó°ÊÓ

The first step in evaluating an iterated integral is to focus on the **inner integral**. Here, our inner integral is \[ \int_{0}^{4} \sqrt{uv} \, du \]which means we are integrating with respect to \(u\) while treating \(v\) as a constant.
To simplify this integral, we use a technique called substitution. By letting \(t = \sqrt{u}\), we convert \(u\) into \(t^2\), and the differential \(du\) becomes \(2t\, dt\).

After substitution, the limits of the integral change accordingly. When \(u = 0\), \(t = \sqrt{0} = 0\), and when \(u = 4\), \(t = \sqrt{4} = 2\). The revised integral becomes:

• \(2v \int_{0}^{2} t^2 \, dt\)

This approach allows us to use simpler functions for integration, making the process more manageable.

The **Substitution Method** is a powerful tool in calculus for finding integrals of complex functions. When faced with a complicated integral, substitution lets us transform it into a more familiar form. In our problem, dealing with \(\sqrt{uv}\) can be tricky directly, hence we use substitution.
The key idea is to replace part of the integral, introducing a new variable. In the inner integral:

• Let \(t = \sqrt{u}\), so \(u = t^2\) and \(du = 2t \, dt\).
• This transforms the integral from a function of \(u\) into one of \(t\), simplifying the process.

Substitution effectively 'reshuffles' the variables to provide something easier to work with. Once the integration is complete in terms of \(t\), convert back to the original variable if necessary.

With the **outer integral**, we finally deal with the function obtained from the inner integration result. After evaluating the inner integral, we've simplified our expression to:

• \(\int_{1}^{4} \frac{16v}{3} \, dv\)

The outer integral involves integrating with respect to \(v\). This is often more straightforward as substitution is usually done in the inner integral.
For our problem, integrating \(\frac{16v}{3}\) with respect to \(v\) yields \(\frac{8v^2}{3}\). Applying the limits from 1 to 4, we find:

• \([\frac{8(4)^2}{3} - \frac{8(1)^2}{3}]\)

Calculating this expression gives us \(40\), which is the final answer to the iterated integral. This step is crucial, as it brings together all previous work to provide the complete solution.