91Ó°ÊÓ

The Power Rule for Integration is a fundamental concept that simplifies taking integrals of functions in the form of a power of a variable. If you have a function of the form \( t^n \), where \( n \) is any real number except -1, the Power Rule states:

• \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \)

In this rule, \( C \) represents the constant of integration, which is typically ignored in definite integrals since the limits of integration provide us with an exact result. This rule is especially convenient in our example since it allows us to integrate terms like \( t^{1/2} \) and \( t^{-1/2} \), transforming them into polynomials by adjusting their exponents. Once the power rule is applied, we have polynomials that are far easier to evaluate over a specified interval.

Evaluating definite integrals involves both finding the antiderivative of a function and then evaluating it at the given limits of integration. For example, to evaluate \( \int_{4}^{9} t^{1/2} \, dt \) from 4 to 9:

• First, find the antiderivative using the Power Rule: \( \frac{2}{3}t^{3/2} \).
• Then, apply the limits: calculate \( \left[ \frac{2}{3}t^{3/2} \right]_{4}^{9} \), which results in \( \frac{2}{3}(9^{3/2} - 4^{3/2}) \).

This gives you the "net area" under the curve described by the function between \( t=4 \) and \( t=9 \). The same process applies to the other part, \( -3\int_{4}^{9} t^{-1/2} \, dt \), leading to finding its antiderivative and then evaluating it within the same bounds. This process allows you to find a definitive numeric value representing the total "accumulation" of the function over that interval, greatly simplifying the understanding of areas under curves.

Simplifying an integrand is a crucial initial step when working with definite integrals as it often reduces a complex expression into a more manageable form. In our original exercise, the expression \( \frac{t-3}{\sqrt{t}} \) can be initially intimidating, but by using algebraic manipulation, we can rewrite it:

• Express \( \frac{t}{\sqrt{t}} - \frac{3}{\sqrt{t}} \) as \( t^{1/2} - 3t^{-1/2} \).

This step smooths the analytical path since dealing with terms like \( t^{1/2} \) and \( t^{-1/2} \) is straightforward with the Power Rule. Simplification makes the integration process faster and less prone to error, setting up the problem in a way that's easier to solve. It reduces complications, ensuring each term aligns with known integration techniques.