91Ó°ÊÓ

The limits of integration are the boundaries within which an integral is evaluated. For the integral \(\int_{0}^{3} \frac{-8}{x \sqrt{x}} dx\), we have limits of integration from 0 to 3. These boundaries are crucial because they determine the area under the function that we are calculating. However, if the integrand (the function being integrated) becomes undefined within these limits, we need to handle it carefully.
In this example, the integrand \(-8x^{-3/2}\) is undefined at \(x = 0\). This is why we consider making the integral an improper integral, by taking a limit as we approach the problematic point, \(x=0\). This helps us assess whether the integral converges to a finite value or diverges to infinity. Handling limits of integration properly is essential when dealing with improper integrals, as incorrect handling can lead to wrong conclusions about the integral's behavior.

An antiderivative of a function is another function whose derivative is the original function. Finding the antiderivative is a crucial step in solving integrals.
For the given integrand \(-8x^{-3/2}\), the antiderivative is computed by reversing the power rule of differentiation. Here, \(-8x^{-3/2}\) becomes \(-8 \frac{x^{-1/2}}{-1/2} = 16 x^{-1/2}\), which can be rewritten as \16 \frac{1}{\sqrt{x}}\.
When we integrate from \(\beta\) to 3, the definite integral becomes \lim_{\beta \to 0^+} 16 \left(\frac{1}{\sqrt{3}} - \frac{1}{\sqrt{\beta}}\right)\. Solving this integral helps us understand the behavior at the boundaries, but we must assess its behavior at these points to determine if it converges or diverges.

Divergence in the context of integrals means that the integral does not settle to a finite value but instead grows without bound. This is a key concept when dealing with improper integrals, especially when an integrand becomes unbounded within the limits of integration.
In our example, as \(\beta \to 0^+\), \frac{1}{\sqrt{\beta}} \to \infty\, leading \(16 \left(\frac{1}{\sqrt{3}} - \frac{1}{\sqrt{\beta}}\right)\) to diverge. Therefore, the integral does not converge to a finite value; it diverges.
It's essential to recognize which parts of the integral contribute to its divergence. Here, the term \(\frac{1}{\sqrt{\beta}}\) dominates as \(\beta \to 0^+\). Recognizing divergence helps us better understand the behavior of functions and their integrals, especially in applied mathematics and calculus.