The area of the region bounded by the graphs of y=x^3 and...

The area of the region bounded by the graphs of $y=x^{3}$ and $y=x$ cannot be found by the single integral $\int_{-1}^{1}\left(x^{3}-x\right) d x .$ Explain why this is so. Use symmetry to write a single integral that does represent the area.

Key Concepts

Area Between Curves

When finding the area enclosed by two curves, it is essential to integrate the absolute difference between the functions (the 'top' function minus the 'bottom' function) over the interval determined by their intersection points. This ensures that all contributions to the area are nonnegative, regardless of which function is above the other over different portions of the interval.

Net Signed Area versus Absolute Area

The integral ?[a,b] (f(x) - g(x)) dx calculates the net signed area, meaning that areas where f(x) < g(x) contribute negatively, potentially canceling out regions where f(x) > g(x). For regions where the roles of the upper and lower curves switch, a single integral without accounting for absolute differences can lead to a cancellation error, as observed in the problem where the integral ? from -1 to 1 (x^3 - x) dx equals zero even though the area between the curves is positive.

Symmetry in Integration

Symmetry can simplify the calculation of areas by reducing the problem to one over a half-interval and then doubling the result. In cases where the region and the functions exhibit symmetric properties relative to the y-axis (or another axis), one can set up an integral over the positive half and multiply by two to find the total area. For instance, splitting the domain for the functions y = x^3 and y = x into [0, 1] (where the order of the functions is consistent) and doubling the integral avoids the cancellation issue present in the full symmetric interval.

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