91Ó°ÊÓ

A double integral is a way to extend the concept of integration, which we usually apply over a single variable, to functions of multiple variables. Specifically, it's used to integrate functions over a two-dimensional area. In our given problem, we aim to approximate the double integral \[ \iint_{R} x \, dA \] where \(R\) is the region bounded by the curves \(y=x^2-1\) and \(y=\frac{1}{1+x^2}\). This involves finding the total "sum" of the function \(x\) over this region, essentially capturing the area-weighted summation of the function's values.

• Setting up the double integral requires identifying the region over which we're integrating.
• We convert the two-dimensional integral into an iterated integral, often expressed as nested integrals.
• The resulting function is integrated with respect to one variable while treating the other as a constant, and then switching to the next variable.

This approach is beneficial for computing areas, volumes, and in this case, understanding the behavior of functions over a defined region.

Determining the intersection of curves is a core step when working with regions in multivariate calculus. In this problem, we need to find the points where the curves \(y = x^2 - 1\) and \(y = \frac{1}{1+x^2}\) meet. This gives us the boundaries for our region \(R\).

• To find intersections, equate the two equations: \(x^2 - 1 = \frac{1}{1+x^2}\).
• Multiply through by \(1 + x^2\) to clear the fraction: \((x^2 - 1)(1 + x^2) = 1\).
• Simplify to a polynomial equation: \(x^4 + x^2 - 2 = 0\).

Using a calculator or software, solve this quartic equation to find \(x\) values. For this problem, these are approximately \(-1\) and \(1\), indicating the region spans from \(x = -1\) to \(x = 1\). This helps set the limits of integration and defines where these functions intersect on the graph.

Various integration techniques are used to evaluate complex integrals, especially when dealing with functions involving polynomials and rational functions, as seen in this problem. The integral to be solved is: \[ \int_{-1}^{1} \left(x^3 - x - \frac{x}{1+x^2}\right) \, dx \] This integral consists of terms that need different approaches.

• Polynomial Terms: The terms \(x^3\) and \(x\) require basic integration rules since they are straightforward polynomials.
• Substitution Method: For the term \(\frac{x}{1+x^2}\), substitution simplifies the integration. By letting \(u = 1+x^2\), \(du = 2x\,dx\), this term transforms into a simpler form.
• Symmetry Considerations: Notably, the integrals of \(x^3\) and \(x\) over symmetric limits, such as \([-1, 1]\), yield zero due to their odd function nature.

Combining these techniques is crucial for solving the entire integral effectively, leading to the conclusion that the entire integral evaluates to zero.