Problem 376 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

The area is approximately 0.5.

Step 1: Identify the Problem

We need to find the area under the curve of the function \(f(x) = x e^{-x^2}\) between \(x = 0\) and \(x = 5\). This involves calculating the definite integral \(\int_{0}^{5} x e^{-x^2} \, dx\).

Step 2: Confirm the Method

To find the area under the graph of \(f(x)\), we use definite integration. Since the function involves an exponential and polynomial part, a substitution method will be needed.

Step 3: Substitution

Let \(u = -x^2\), then \(du = -2x \, dx\). Rearrange the equation to get \(x \, dx = -\frac{1}{2} \, du\). This substitution simplifies the integral.

Step 4: Change the Limits of Integration

When \(x = 0\), \(u = 0\). When \(x = 5\), \(u = -25\). Substitute these into the integral, changing the limits accordingly.

Step 5: Write the New Integral

With the substitution and limits, the integral becomes \(\int_{0}^{-25} e^{u} \left(-\frac{1}{2}\right) \, du\). Thus, it simplifies to \(-\frac{1}{2} \int_{0}^{-25} e^{u} \, du\).

Step 6: Integrate

The integral of \(e^{u}\) with respect to \(u\) is simply \(e^{u}\). Therefore, \(-\frac{1}{2} \int_{0}^{-25} e^{u} \, du\) becomes \(-\frac{1}{2} [e^{u}]_{0}^{-25}\).

Step 7: Evaluate the Definite Integral

Evaluate \([-\frac{1}{2} (e^{-25} - e^{0})]\). Calculate \(e^{-25}\) and \(e^{0} = 1\), leading to \(-\frac{1}{2} (e^{-25} - 1) \).

Step 8: Approximate the Answer

Since \(e^{-25}\) is a very small number close to zero, the approximate value of the integral is \(-\frac{1}{2} (0 - 1) = \frac{1}{2}\).

Key Concepts

Substitution Method

The substitution method is an essential technique used in integration, particularly when dealing with complex functions involving combinations of exponential and trigonometric parts. It simplifies the integration process by replacing the variable of integration with a new variable.

To employ the substitution method, follow these steps:

• Identify a substitution that simplifies the integral. This often involves substituting a more complicated expression with a single variable, typically denoted as \(u\).
• Derive the differential for the new variable (e.g. \(du\)) and express the original differential (e.g., \(dx\)) in terms of \(du\).
• Adjust the limits of integration if dealing with a definite integral, by expressing them in terms of the new variable.

Using a proper substitution method allows you to transform a difficult integral into a more manageable form, making evaluation easier. In our exercise, the substitution \(u = -x^2\) simplifies the integral of the function \(f(x) = x e^{-x^2}\). This alteration results in a new integral, which is easier to evaluate.

Exponential Function

Exponential functions are a vital part of mathematical analysis, characterized by having a constant base raised to a variable exponent, often expressed as \(e^{x}\), where \(e\) is Euler's number. Key properties of exponential functions include:

• The function's rate of change is proportional to its value.
• Exponential functions can model real-world phenomena such as population growth and radioactive decay.

In the realm of calculus, exponential functions frequently present challenges due to their non-linear properties.

For integrals involving exponential functions, recognizing when to use properties like \(e^{a+b} = e^{a}e^{b}\) can simplify calculations. In our problem, the function involves \(e^{-x^2}\), an exponential decay function, part of a broader category useful in describing the "rate of decay" of a quantity.

When integrating functions involving exponentials, understanding these characteristics is crucial. The exponential term \(e^{-x^2}\) contributes significantly to the behavior of the function \(f(x) = x e^{-x^2}\), especially when calculating area under the curve through integration.

Area Under the Curve

Finding the area under a curve is a fundamental concept in calculus, often equating to calculating a definite integral. This is especially relevant when determining physical quantities, such as total displacement over a time interval when provided with a velocity function.

The process involves computing the integral of a function over a given interval, which provides a numerical "area" between the graph of a function and a specified axis.

• The definite integral \(\int_{a}^{b} f(x) \, dx\) represents the accumulated area under the curve \(f(x)\) from \(x=a\) to \(x=b\).
• If the function is above the \(x\)-axis, the integral gives the exact area. If below, it returns a negative value, indicating area measured in the opposite direction.

In the provided exercise, the integral \(\int_{0}^{5} x e^{-x^2} \, dx \) finds the area under the curve of the function \(f(x)\) from \(x=0\) to \(x=5\). The accurate application of definite integration reveals how the exponential decay and polynomial interplay define the exact space between the function and the axis.