91Ó°ÊÓ

A definite integral is a fundamental concept in calculus, particularly used to calculate the signed area under a curve within a specific interval. In this context, we're using definite integrals to find the volume of a solid of revolution, which is formed by rotating a bounded area around an axis—in our case, the x-axis.
The process involves setting up an integral function that represents the shape of the solid formed. Typically, this is expressed in the form:

• \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \],

which helps determine the volume of the solid.
In our example, we have the function \( y = \frac{2}{\sqrt{x}} \). By squaring it and integrating over the interval [4, 9], the definite integral allows us to compute the precise volume by taking into account the entire bounded region under the curve.

In our exercise, logarithmic properties play an essential role in simplifying expressions arising from integration. After performing an integration that results in a logarithmic function, we can use these properties to make the expression easier to handle.
One key property of logarithms used here is \( \ln{a} - \ln{b} = \ln{\frac{a}{b}} \). This property allows us to combine or simplify logarithmic differences into a single expression.
In the solution, we first encounter \( 4 \ln{9} - 4 \ln{4} \), and using the aforementioned property, it is rewritten as \( 4 \ln{\frac{9}{4}} \). These operations are vital because they aid in reducing complex expressions, making integral calculations more manageable, and ensuring we have the most compact form for further computation or interpretation.

Integration of a function is the process of finding its antiderivative or the integral itself. In calculus, function integration lets us determine areas under curves, solve differential equations, and find volumes of solids, among other applications.
In the context of this exercise, we're integrating the function \( \frac{4}{x} \) after converting \( \left(\frac{2}{\sqrt{x}}\right)^2 \) into a simpler form using basic algebraic manipulation. The integration of \( \frac{1}{x} \) is a standard case resulting in the natural logarithm function \( \ln{|x|} \).
By multiplying the integral by 4 due to our simplification process, we evaluate \( 4 \ln{|x|} \) over the interval [4, 9]. Finally, function integration gives us a precise mathematical framework that, by following specific rules and techniques, helps transform complex real-world problems like volume calculation into solvable mathematical procedures.