Question

Find the area under the graph of the function \(f(x)=x e^{-x^{2}}\) between \(x=0\) and \(x=5\).

Short answer

The area under the graph from \(x=0\) to \(x=5\) is approximately 0.5.

Step-by-step solution

Identify the Integral

To find the area under the curve of the function, we need to compute the definite integral of the function. For the given function \(f(x) = x e^{-x^2}\), the area under the graph between \(x=0\) and \(x=5\) can be found by evaluating the integral \(\int_{0}^{5} x e^{-x^2} \, dx\).

Substitute and Use Integration by Parts

Notice that the integral \(\int x e^{-x^2} \, dx\) can be solved using substitution and integration by parts. Set \(u = -x^2\), then \(du = -2x \, dx\), or \(x \, dx = -\frac{1}{2} du\). The integral becomes:\[ -\frac{1}{2} \int e^{u} \, du \]},

Solve the Integral after Substitution

The integral \(-\frac{1}{2} \int e^{u} \, du \) evaluates to \(-\frac{1}{2} e^{u} + C\). Substitute \(u = -x^2\) back into the expression to write it in terms of \(x\):\[ = -\frac{1}{2} e^{-x^2} + C \]

Evaluate the Definite Integral

Evaluate the definite integral from \(0\) to \(5\):\[ \int_{0}^{5} x e^{-x^2} \, dx = \left[- \frac{1}{2} e^{-x^2} \right]_0^5 \] Substitute the limits:\[ \left[- \frac{1}{2} e^{-5^2} \right] - \left[- \frac{1}{2} e^{-0^2} \right] = \left[- \frac{1}{2} e^{-25} \right] - \left[- \frac{1}{2} e^{0} \right] \]

Simplify and Compute the Result

Simplify the expression:\[ = -\frac{1}{2} e^{-25} + \frac{1}{2} e^{0} = \frac{1}{2} - \frac{1}{2} e^{-25} \]This expression represents the numerical area under the curve from \(x=0\) to \(x=5\).

Calculate Numerically

Calculate \(\frac{1}{2} - \frac{1}{2} e^{-25}\). Since \(e^{-25}\) is very close to zero, this simplifies to:\[ \frac{1}{2} - \frac{1}{2}(0) = \frac{1}{2} \]Thus, the area is approximately \(0.5\).

Key concepts

Definite Integral

To find the area under a curve, we use a concept called the definite integral. In essence, a definite integral represents the total accumulation of values of a function over a specific interval. In our example, we want to find the area under the function \( f(x) = x e^{-x^2} \) from \( x = 0 \) to \( x = 5 \). This is expressed as \( \int_{0}^{5} x e^{-x^2} \: dx \).

Definite integrals can be thought of as the signed area between the curve and the x-axis over a defined range. The result gives the net area, which can be positive or negative depending on the function's position relative to the x-axis.

• The integral symbol \( \int \) indicates integration across an interval.
• The limits of integration, 0 and 5 in this case, specify the interval boundaries.

Evaluating a definite integral requires calculating an antiderivative and then applying the Fundamental Theorem of Calculus to find the function values at the interval boundaries.

Integration by Parts

Integration by parts is a crucial technique often used when simple substitution won't suffice. It's particularly helpful when dealing with products of functions like \( x e^{-x^2} \). Integration by parts is derived from the product rule for differentiation and follows the formula:

\[ \int u \, dv = uv - \int v \, du \]

In practice, we choose parts of the integrand as \( u \) and \( dv \), differentiate \( u \) to get \( du \), and integrate \( dv \) to find \( v \). This technique effectively reduces the complexity of the integral.

• It's useful for integrating products of polynomial and exponential functions, like our example.
• Choosing \( u \) and \( dv \) wisely can simplify the solution.

For the integral \( \int x e^{-x^2} \, dx \), substitution is first used to make it manageable, and then integration by parts can further address integrals that substitution alone can't resolve.

Substitution

The substitution method makes integration easier by transforming the integrand into a simpler form. It's based on the change of variables technique, similar to the reverse of differentiation.

• Identify a function within the integral to simplify. Typically, this is part of a composite function.
• Express the integral in terms of a new variable, simplifying the computation.
• Remember to change the integration limits if evaluating a definite integral.

For \( \int_{0}^{5} x e^{-x^2} \, dx \), setting \( u = -x^2 \) allows us to rewrite the differential as \( du = -2x \, dx \), simplifying the integral to \( -\frac{1}{2} \int e^{u} \, du \).

Substitution is invaluable for integrals involving composite functions, where it streamlines the integration process significantly.

Numerical Integration

When it is exceptionally challenging to find an exact antiderivative, numerical integration comes into play. It approximates the value of a definite integral without needing an analytical solution. This is beneficial for complex or unsolvable integrals.

There are several methods for numerical integration:

• The Trapezoidal Rule and Simpson's Rule are common techniques, using simple geometric shapes to approximate area.
• They provide estimates that improve with finer subdivisions of the interval.

Although for \( \int_{0}^{5} x e^{-x^2} \, dx \), an analytical solution exists, using \( e^{-25} \), which is close to zero, allows for quick estimation of the integral, resulting in \( 0.5 \).

Numerical techniques are essential tools, especially when tackling integrals difficult to solve using traditional methods.