Question

Compute the integral of $f(x)=x{e}^{-x^2}$ and find the smallest value of $N$ such that the area under the graph $f(x)=x{e}^{-x^2}$ between $x=N$ and $x=N+10$ is, at most, $0.01$

Short answer

The smallest value of $N$ is approximately 3.4.

Step-by-step solution

Set Up the Integral

We want to find the integral of the function $f(x)=x{e}^{-x^2}$. The integral can be expressed as $\int x{e}^{-x^2}dx$.

Use Substitution Method

Let us use substitution. Set $u=-x^2$, then $du=-2xdx$, or $xdx=-\frac{1}{2}du$. Substitute for $u$ in the integral: $\int x{e}^{-x^2}dx=\int -\frac{1}{2}{e}^{u}du$.

Integrate with respect to $u$

We now integrate $\int -\frac{1}{2}{e}^{u}du$. This results in $-\frac{1}{2}\int e^{u}du=-\frac{1}{2}{e}^u+C$.

Substitute back to the original variable

Substitute $u=-x^2$ back into the function: $-\frac{1}{2}{e}^u+C=-\frac{1}{2}{e}^{-x^2}+C$. This gives us the indefinite integral $\int x{e}^{-x^2}dx=-\frac{1}{2}{e}^{-x^2}+C$.

Integrate over the interval $[N,N+10]$

To find the area between $x=N$ and $x=N+10$, evaluate the definite integral: $-\frac{1}{2}{e}^{-x^2}{|}_N^{N+10}=-\frac{1}{2}{e}^{-(N+10{)}^2}+\frac{1}{2}{e}^{-N^2}$.

Set the condition for the area

We need this area to be at most 0.01, so set up the inequality: $-\frac{1}{2}{e}^{-(N+10{)}^2}+\frac{1}{2}{e}^{-N^2}\le 0.01$.

Simplify and Solve the Inequality

Simplify the inequality: $e^{-N^2}-e^{-(N+10{)}^2}\le 0.02$. Solving this generally requires numerical methods to find the smallest value of $N$.

Numerical Approximation

Using numerical or graphing tools, estimate the smallest $N$. By checking values starting from a high number, you'll find $N\approx 3.4$ satisfies the inequality.

Key concepts

Indefinite Integral

An indefinite integral, also known as an antiderivative, is a function that reverses the process of differentiation. It represents a family of functions whose derivative is the integrand, the function inside the integral. For the function given in this problem, the indefinite integral is written as $\int x{e}^{-x^2}dx$. The process of finding the indefinite integral includes using integration techniques such as substitution.
Indefinite integrals are crucial because they help us understand the general form of the solution before deciding on particular values, such as initial conditions or specific boundaries in definite integrals. Once the indefinite integral is computed, the result usually includes a constant $C$, which accounts for all possible vertical shifts of the antiderivative. In our example, the result $-\frac{1}{2}{e}^{-x^2}+C$ entails the general solution for $\int x{e}^{-x^2}dx$.
This constant $C$ is removed when evaluating definite integrals, as definite integrals provide a specific numerical value representing the area under the curve between given bounds.

Substitution Method

The substitution method is a technique used to simplify integration by making it easier to evaluate. It is similar to the chain rule in differentiation, but used in reverse. This method involves changing the variable of integration from $x$ to another variable $u$. In this process, we make a substitution that transforms the integral into a basic form that is easier to solve.
For our function $f(x)=x{e}^{-x^2}$, we set $u=-x^2$, which translates to $du=-2xdx$. Consequently, we find that $xdx=-\frac{1}{2}du$. This transforms our integral into $\int -\frac{1}{2}{e}^{u}du$, which is a more straightforward form to integrate. After performing the integration, we substitute back from the variable $u$ to $x$ to obtain the final form of the indefinite integral.
Substitution is a powerful tool because it allows us to tackle more complex integrals by reducing them to simpler, more familiar forms, making the overall integration process much more manageable.

Numerical Approximation

Numerical approximation is a method used when an exact analytical solution to an integral is difficult or impossible to find. It involves using numerical methods to estimate the value of definite integrals. In our exercise, once the inequality $e^{-N^2}-e^{-(N+10{)}^2}\le 0.02$ was established, direct analytical solving was not straightforward. Thus, numerical approximation comes into play.
Several numerical methods can be applied, such as the trapezoidal rule, Simpson's rule, or using computational software or graphing calculators. These tools enable you to find values of $N$ by iteratively checking different possibilities, ensuring that the resultant area equation satisfies the inequality condition.
In our case, numerical methods estimated the smallest value of $N$ to be approximately 3.4. Numerical approximation is especially useful in real-world applications where precise calculations are necessary, but the functions involved do not allow for simple integration.

Exponential Function

The exponential function is a mathematical function denoted as $e^x$, where $e$ is a constant approximately equal to 2.71828. This function is central to calculus, appearing frequently in growth models and decay processes, as well as in complex-valued functions and calculus applications.
In the given problem, the function $f(x)=x{e}^{-x^2}$ includes an exponential component with a negative exponent, which indicates exponential decay. The nature of $e^{-x^2}$ means that as $x$ increases, $e^{-x^2}$ rapidly approaches zero, contributing to the behavior of the overall function. This exponential feature is essential because it significantly affects the integral's value over specified intervals. For the problem at hand, understanding the nature and behavior of the exponential component is crucial to evaluating and approximating the integral correctly.
The exponential function's unique properties make it vital across various fields, ranging from natural phenomena modeling to financial calculations, providing deep insights into patterns of change and growth.