91Ó°ÊÓ

Integration is a fundamental tool in calculus used to find areas under curves, accumulate quantities, or solve differential equations. In this exercise, we start by expanding a function that involves square roots and polynomials. Understanding and applying integration techniques is crucial to simplify and evaluate integrals effectively.
To solve the integral \( \int_{2}^{0} \sqrt{x}(x+1)(x-2) \, dx \), we first need to simplify it. This involves expanding the polynomial multiplication \((x+1)(x-2)\), which gives \(x^2 - x\).

• Next, using the distributive property, multiply the factor \(\sqrt{x}\) through the simplified polynomial.
• This results in an expression in terms of powers of \(x\) that are easier to integrate: \(x^{\frac{3}{2}} - x^{\frac{1}{2}}\).

These steps highlight the integration technique known as polynomial expansion. Simplifying the integrand allows us to apply standard integration rules effectively by transforming complex terms into simpler polynomials.

The Fundamental Theorem of Calculus is a cornerstone in understanding the relationship between differentiation and integration. It bridges the processes of finding antiderivatives and calculating definite integrals.
The theorem states that if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then the definite integral over \([a, b]\) is given by:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
In our exercise, after finding the antiderivative of \( x^{\frac{3}{2}} - x^{\frac{1}{2}} \), we apply this theorem.

• Evaluate \( F \) at the upper bound of the interval, \(2\), and at the lower bound, \(0\).
• This involves substituting these limits into the antiderivative: \( F(x) = \frac{2}{5}x^{\frac{5}{2}} - \frac{2}{3}x^{\frac{3}{2}} \).
• Subtract the values to determine the integral: \( F(0) - F(2) = 0 - (-1.12097) \).

The result provides the area under the curve from \( 2 \) to \( 0 \), showing the practical utility of the theorem in calculating areas and integrals.

Polynomial expansion is a technique used to simplify expressions that involve products of sums or differences. Here, it helps us rewrite the function into a form that is easier to integrate.
At the start of the exercise, \( f(x) = \sqrt{x}(x+1)(x-2) \) is expanded.

• First, recognize \( (x+1)(x-2) = x^2 - x - 2 \), using distributive property over multiplication.
• Then, introduce \( \sqrt{x} = x^{\frac{1}{2}} \).
• This results in \( x^{\frac{5}{2}} - 2x^{\frac{3}{2}} - x^{\frac{3}{2}} \), after distributing \( x^{\frac{1}{2}} \).
• Simplify the expression to \( x^{\frac{3}{2}} - x^{\frac{1}{2}} \), ensuring terms are ordered by the power of \( x \).

This expansion process simplifies integrals, making them solvable using straightforward integration techniques. Understanding polynomial expansion allows us to navigate complex integrals by breaking them into manageable parts.