Consider the function f(x)=4 x e^-x^2. (a) Find the signed...

Consider the function $f(x)=4 x e^{-x^{2}}$.
(a) Find the signed area of the region between the graph of $f(x)$ and the $x$ -axis on $[-1,2]$ shown here:
(b) Find the absolute area of the same region.

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Key Concepts

Signed Area vs. Absolute Area

The signed area counts regions below the x-axis as negative and those above as positive, reflecting the net offset between them. In contrast, absolute area refers to the total area irrespective of sign, obtained by integrating the absolute value of the function, which often requires partitioning the domain based on where the function changes sign.

Substitution Method

The substitution method is a technique used in integration to simplify an integral by changing variables. It is particularly useful when the integrand involves a composite function, allowing the integral to be rewritten in a form that is easier to evaluate using known antiderivatives.

Definite Integral

The definite integral is a fundamental concept in calculus that computes the net accumulation of a quantity, such as area under a curve, between two specified limits. It represents the signed area, where regions above the x-axis contribute positively and regions below contribute negatively.

Fundamental Theorem of Calculus

This theorem bridges differentiation and integration by stating that if a function is continuous on a closed interval, then its definite integral can be computed using an antiderivative. It provides a practical way to evaluate definite integrals as the difference in the antiderivative evaluated at the endpoints.

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