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The power rule for integration is a fundamental technique used in calculus to find the integral of expressions involving powers of variables. This rule is similar to the power rule for differentiation, but reversed. If you can master this rule, integrating basic power functions becomes quite straightforward.

To apply the power rule for integration, remember the formula:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \),

where \( n eq -1 \) and \( C \) is the constant of integration.
This tells us that to integrate a term like \( x^n \), you should:

• Add one to the exponent \( n \).
• Divide by the new exponent.

For example, for \( 3x^{1/2} \), we apply the rule and find \( 2x^{3/2} \) because when \( n = 1/2 \), the integral becomes \( \frac{x^{3/2}}{3/2} \), and multiplied by 3 gives \( 2x^{3/2} \). Similarly, for \( 2x^{-1/2} \), the integral results in \( 4x^{1/2} \) after applying the power rule and multiplying by 2.

The key takeaway is to always adjust the exponent, divide by the new exponent, and remember the constant of integration for indefinite integrals.

Definite integrals allow us to find the net area under a curve within specified boundaries—we call these boundaries the limits of integration. The notation for definite integrals includes these limits, written as \( \int_{a}^{b} \), where \( a \) is the lower limit and \( b \) is the upper limit.

To evaluate a definite integral, follow these steps:

• First, find the antiderivative of the function. This involves integrating as if there were no bounds, using techniques like the power rule.
• Next, apply the limits of integration. This involves substituting the upper limit and then the lower limit into the antiderivative.
• Finally, subtract the result of the lower limit substitution from that of the upper limit. This calculation gives you the definitive value of the integral, representing the net area between the function and the x-axis over the interval \([a, b]\).

In our case, integrate \( 2x^{3/2} \) and \( 4x^{1/2} \) separately, substitute the boundaries \( x=9 \) and \( x=1 \), and then perform subtraction to combine the results. This combination of the two calculations is often seen in step-by-step calculus solutions, highlighting the importance of methodically applying limits.

Integral calculus is a major area of calculus that deals with integrals and the concept of accumulation. It's integral to understanding changes across various disciplines, from areas under curves to physical quantities like distance and volume.

The fundamental principle behind integral calculus is the idea of finding integrals, either indefinite or definite:

• Indefinite Integrals: These are general forms of antiderivatives, providing a broad expression for a family of functions without specific evaluation limits.
• Definite Integrals: These represent a numerical value defined by a specific range, or interval, and are often used to calculate areas under curves or between two points.

Integral calculus employs several rules and techniques, with the power rule being one of the simplest yet powerful tools at a student's disposal. These methods allow integration of different forms of functions, transforming our understanding of complex dynamic systems.

In practice, integral calculus extends to solving real-world problems: from engineering and physics calculations for determining work done by forces, to economics for determining consumer and producer surplus. Understanding the implementation and computation of integrals, especially the use of definite integrals, unlocks a wealth of practical applications.