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Integration by substitution is a popular method to simplify complex integrals. It involves replacing part of the integral with a new variable to make integration easier. In our case, we have the function \(f(x) = x e^{x^2}\). This integrand suggests substitution because the derivative \(d(x^2) = 2x \, dx\) appears in the original expression.
To proceed, we choose the substitution \(u = x^2\). This transformation changes the differential \(dx\) into a form that aligns with the derivative of the function \(x e^{x^2}\). With \(du = 2x \, dx\), we can rearrange to get \(\frac{1}{2} \, du = x \, dx\). Therefore, the integral becomes simpler:

• The original limits \(x = 0\) to \(x = 2\) transform to \(u = 0\) to \(u = 4\).
• The integral \(\int_{0}^{2} x e^{x^2} \, dx\) becomes \(\frac{1}{2} \int_{0}^{4} e^{u} \, du\).

Simplifying the integrand using substitution significantly eases the process of finding the area under the curve.

Exponential functions are functions of the form \(e^u\), where \(e\) is the base of the natural logarithm. These functions are important in calculus because they have straightforward derivatives and integrals.
In the integral \(\int_{0}^{4} e^u \, du\), the exponential function \(e^u\) is very easy to integrate. The integral of \(e^u\) with respect to \(u\) is simply \(e^u\), showcasing another useful property of the exponential function.
The simplicity of this operation is a key reason why transforming an integrand into an exponential function through substitution can be particularly effective. It leads to:

• Simplifying the integration process.
• Allowing straightforward evaluation by substitution.

This is why exponential functions are a common feature in integrating functions in calculus.

The concept of the area under a curve is fundamental in calculus, especially when interpreting integrals. When you compute the integral of a function over an interval, you essentially calculate the net area between the curve and the x-axis.
For the function \(f(x) = x e^{x^2}\), finding the area between \(x = 0\) and \(x = 2\) involves evaluating the definite integral \(\int_{0}^{2} x e^{x^2} \, dx\). Through integration by substitution, we found this area to be \(\frac{1}{2} (e^4 - 1)\).
This expression represents the exact area under the curve of \(f(x)\) within the given bounds. In the context of applications:

• This calculation helps in modeling real-life scenarios where understanding total values from continuously varying quantities is necessary.
• In practice, it can be used to compute things like accumulated interest, population growth or decay, or physical quantities like distance or volume.

Understanding how to calculate the area under a curve using integrals is thus a powerful tool in both mathematics and its applications.