91Ó°ÊÓ

The Power Rule of Integration is a fundamental tool for finding antiderivatives, especially when dealing with polynomial expressions. It's closely tied to the Power Rule for differentiation, but works in reverse. If you're given a term in the form of \(x^n\), the Power Rule allows you to find its integral by adding 1 to the exponent \(n\) and then dividing by the new exponent. This is summarized as:

• For \(x^n\), the integral is \(\frac{x^{n+1}}{n+1}\), provided \(n eq -1\).

When applied to each term in a polynomial, it simplifies the integration process significantly. In our exercise, after expanding \((x-6)^2\) into \(x^2 - 12x + 36\), we applied the Power Rule to each term:

• The integral of \(x^2\) becomes \(\frac{x^3}{3}\).
• The integral of \(-12x\) becomes \(-6x^2\).
• The integral of \(36\) becomes \(36x\).

Using the Power Rule effectively, we transform the polynomial into an easily manageable form for further processing.

The Fundamental Theorem of Calculus is a pivotal result in mathematics that bridges the concepts of differentiation and integration. It comprises two parts that together show how we can compute the definite integral of a function using its antiderivative. The key here is understanding its application in definite integrals:

• First, find the antiderivative of the function.
• Then, evaluate this antiderivative at the upper and lower limits of integration.
• Subtract the value at the lower limit from the value at the upper limit.

In our exercise, once we derived the antiderivative \(\frac{x^3}{3} - 6x^2 + 36x\), we applied the Fundamental Theorem:

• We evaluated \(\left(\frac{x^3}{3} - 6x^2 + 36x\right)\) at \(x = 4\) and \(x = -7\).
• Subtracting the two results gives us the definite integral's value.

This procedure not only finds the area under the curve between the given limits but also makes integration a powerful tool for solving practical problems.

Polynomial Expansion is a technique used to express a polynomial that is raised to a power, such as \((x-6)^2\), in a simpler form. The expansion allows for the polynomial to be integrated term by term more easily.

• This process involves expanding the expression by applying the distributive property of multiplication.
• For example, \((x-6)^2\) expands to \(x^2 - 12x + 36\).

This expanded form is crucial for applying integration rules, such as the Power Rule. In our step-by-step solution:

• We first expanded \((x-6)^2\) to simplify the integration process.
• This led to easier management of each term during integration.

Understanding polynomial expansion is essential, as it provides a foundation for handling more complex integrals by breaking them down into simpler, more manageable parts. By mastering expansion techniques, one can effectively tackle a vast array of integrals.