Question

Evaluate the given double integral by changing it to an iterated integral. \(\iint_{S}\left(x^{2}-x y\right) d A ; S\) is the region between \(y=x\) and \(y=3 x-x^{2}\)

Short answer

The value of the double integral is computed by evaluating the iterated integral.

Step-by-step solution

Determine the Region of Integration

First, we need to sketch or determine the region of integration, \( S \), which is bounded by the curves \( y = x \) and \( y = 3x - x^2 \). These curves intersect when \( y = x = 3x - x^2 \). Solving the equation \( x = 3x - x^2 \) gives us the intersection points. Simplifying, we get \( x^2 - 2x = 0 \), or \( x(x-2) = 0 \), so \( x = 0 \) and \( x = 2 \). Therefore, the region of integration is bounded by these intersections from \( x = 0 \) to \( x = 2 \).

Set Up the Iterated Integral

Now that we know the bounds, we convert the double integral to iterated integrals. For each \( x \) from 0 to 2, \( y \) goes from \( y = x \) to \( y = 3x - x^2 \). Thus, the integral becomes: \[ \int_{0}^{2} \int_{x}^{3x-x^2} (x^2 - xy) \ dy \ dx \]

Integrate with Respect to \( y \)

First, we integrate the inner integral with respect to \( y \):\[ \int_{x}^{3x-x^2} (x^2 - xy) \ dy = \left[ x^2y - \frac{xy^2}{2} \right]_{x}^{3x-x^2} \]Compute this by plugging in the limits for \( y \):\[ = \left( x^2(3x-x^2) - \frac{x(3x-x^2)^2}{2} \right) - \left( x^2(x) - \frac{x(x)^2}{2} \right) \]

Simplify the Expression

Simplifying, we have:\[ = x^2(3x-x^2) - \frac{x(3x-x^2)^2}{2} - (x^3 - \frac{x^3}{2}) \]Expand each term and simplify thoroughly:\[ = (3x^3 - x^4) - \frac{x}{2}(9x^2 - 6x^3 + x^4) - (x^3 - \frac{x^3}{2}) \]Simplifying each expression thoroughly provides a polynomial in \( x \) to integrate next.

Integrate with Respect to \( x \)

Now, integrate the resulting expression with respect to \( x \) from 0 to 2:\[ \int_{0}^{2} (.....) \ dx \]After integrating term by term, find the difference when evaluated from \( x=0 \) to \( x=2 \).

Solve and Simplify

Perform the calculations for definite integration over each evaluated polynomial and simplify to find the final answer.

Key concepts

Region of Integration

The "region of integration" is a crucial part of evaluating any double integral. It defines the specific area over which you are integrating, usually denoted as a region in the plane, represented by two functions or boundaries. In our exercise, the region of integration, \( S \), is the area enclosed between the curves \( y = x \) and \( y = 3x - x^2 \). To pin down this region, we solve for the intersection points of these curves. By equating \( y = x \) and \( y = 3x - x^2 \), we find where they intersect: solving \( x = 3x - x^2 \) ultimately provides \( x = 0 \) and \( x = 2 \) as the points of intersection. The region of integration is thus the area between these curves from \( x = 0 \) to \( x = 2 \). Once the region is identified, we can proceed to set up our iterated integral.

Iterated Integral

An "iterated integral" is essentially a way to simplify a double integral by breaking it down into two separate integrals, one inside the other. This is achieved by iteratively integrating over one variable at a time. In this exercise, after determining the region of integration, we reformat the double integral into an iterated form to make computation manageable. By setting up our integral as \( \int_{0}^{2} \int_{x}^{3x-x^2} (x^2 - xy) \, dy \, dx \), we first integrate with respect to \( y \), treating \( x \) as a constant. The altered bounds \( y = x \) to \( y = 3x - x^2 \) reflect this change. Once the inner integral with respect to \( y \) is resolved, the next step is to handle the outer integral, with respect to \( x \). This methodical iteration ensures each variable is addressed individually, simplifying the complex process of double integration.

Definite Integration

"Definite integration" takes an integral and evaluates it within a set boundary, yielding a numerical output. In the context of double integrals, this implies evaluating the integral over our predefined region of integration. Once our iterated integral is established, we first undertake definite integration with respect to \( y \), using the specified bounds. Specifically, starting from \( y = x \) to \( y = 3x - x^2 \). The resulting expression is then subjected to definite integration with respect to \( x \), using the bounds from 0 to 2. Calculating the integral's exact value over these definitive intervals is a major step in capturing the fully integrated effect over the entire region, yielding a comprehensive understanding of how the function behaves over \( S \).

Polynomial Integration

"Polynomial integration" refers to the integration of polynomial expressions, where each term of the polynomial is integrated separately. This is typically straightforward, involving applying basic antiderivative rules. In this exercise, once the iterated integral is simplified, the expression becomes a polynomial in terms of \( x \). Specifically, after substituting the bounds and simplifying the expression, we gather a polynomial which might include terms like \( 3x^3 - x^4 \) and \( -x^3 + \frac{x^3}{2} \). Each term's antiderivative is computed by increasing the power by one and dividing by the new power. After obtaining the antiderivatives of these polynomial terms, the definite integration is performed. Evaluating at the bounds, 0 and 2, and subtracting provides the complete solution to the original double integral problem. This approach demonstrates the process of integrating polynomials systematically within the context of iterated integrals.