Question

\(\int_{0}^{2} x^{3} \sqrt{\left(2 x-x^{2}\right)} d x=\)

Short answer

The integral evaluates to zero due to symmetric boundaries.

Step-by-step solution

Recognize the Problem Type

The given problem is a definite integral problem over a closed interval [0, 2]. It involves a function that is a product of a polynomial and a square root expression.

Simplify the Square Root Expression

The expression inside the square root can be simplified. Notice that \(2x - x^2 = x(2-x)\). Thus, \( \sqrt{2x - x^2} = \sqrt{x(2-x)} = \sqrt{x}\sqrt{2-x} \).

Use Substitution to Simplify the Integral

Let's use a substitution to simplify the integral. Let \(u = 2x - x^2\). Then, \(du = (2 - 2x) dx = 2(1-x) dx\). Rearrange the equation for \(dx\) so \(dx = \frac{du}{2(1-x)}\).

Change Limits of Integration

Calculate the new limits of integration. When \(x = 0\), \(u = 2 imes 0 - 0^2 = 0\). When \(x = 2\), \(u = 2 imes 2 - 2^2 = 0\). However, we must account for if the integral itself evaluates to zero since the limits indicate this directly.

Evaluate the Integral

Rewriting the integral with the substitution, it becomes \(\int x^{3/2} (2-x)^{1/2} \, dx\) which simplifies back to fundamental terms as \(\int u^{1/2} \, du\) with corrected bounds reflecting potential zero estimate, leading to the realization this symmetry applies given boundaries.

Interpret and Conclude

Given the indication that bounds equate, mathematical symmetry illustrates the integral evaluates directly at zero, confirming the product fulfilling the criteria within bounded zero.

Key concepts

Polynomial Integration

Polynomial integration often involves integrating functions that are composed of polynomials. A polynomial is an expression consisting of constants, variables, and non-negative integer exponents of variables. For example, in the original exercise, the term \(x^3\) is a polynomial.When integrating polynomials, we apply the power rule for integration: increase the exponent by one, and divide by the new exponent.- For a polynomial of the form \( x^n \), the integral is \( \frac{x^{n+1}}{n+1} + C \), where \(C\) is the constant of integration.In definite integrals, such as in this exercise, the result is calculated over a specific interval, implying no integration constant is added. Instead, we evaluate the antiderivative at both bounds of the interval and subtract these values to find the area under the curve.To sum up, understanding polynomial integration is crucial, as it builds foundational skills for tackling more complex integrals.

Integration Techniques

Integration is a core part of calculus, dedicated to finding the antiderivative or evaluating the area under a curve. Different functions require different techniques to integrate. Basic functions use straightforward methods, but more complex functions, such as products of polynomials and square roots, might require advanced techniques.- **Simplifying Expressions**: Before integrating, simplify the expression if possible. In the given exercise, we simplified \( \sqrt{2x - x^2} \) to \( \sqrt{x(2-x)} \), breaking it down into \( \sqrt{x} \cdot \sqrt{2-x} \).- **Substitution**: This is a common technique used when a function can be rewritten in terms of another substitute variable. By substituting, you simplify the expression, making it easier to integrate.Each problem demands a careful analysis to choose the most suitable technique. For some, this may include integration by parts or recognizing specific patterns, but the primary goal is always to simplify the problem to something more manageable.

Substitution Method

The substitution method is a powerful and essential technique in integration. It involves changing variables to simplify the integration process. This method is often referred to as "u-substitution", and is particularly useful when dealing with composite functions.Here’s how it typically works:1. **Select the Substitution**: In our exercise, the expression \(2x - x^2\) was chosen as \(u\). This step often requires insight into the problem, seeking expressions which, when differentiated, streamline the integral.2. **Differentiate**: Find \(du\). In the example, \(du = (2 - 2x) dx\) was derived, leading to \(dx = \frac{du}{2(1-x)}\).3. **Change the Integral Bounds**: Substitute the limits of integration. For \(x=0\) and \(x=2\), in our function, both gave \(u=0\), indicating that careful interpretation of limits is crucial, sometimes leading to a zero result or symmetry insights.4. **Integrate and Back Substitute**: Once the substitution has simplified the integral form, solve it using basic integration rules, replacing \(u\) back with the original variables if necessary.Substitution may often look complex at first glance, but with practice, it simplifies problems that could otherwise be very challenging.