Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ \begin{aligned} &y=x e^{-x^{2}}, y=0, x=0, x=1\\\ &\begin{gathered} 51 / 3 \end{gathered} \end{aligned} $$

Step-by-Step Solution

Answer

The region is bounded by \(y=x e^{-x^{2}}\), the x-axis, and the lines x=0 and x=1. The area of this region is \(0.5 - 0.5/e\)

Step 1: Sketch the graphs

The curves are \(y=x e^{-x^{2}}\), a bell-shaped curve with peak at x=0 and asymptotic to x axis, \(x=0\) and \(x=1\) are vertical lines, and \(y=0\) describes the x-axis. Carefully plot these on the same set of axes to identify the bounded region.

Step 2: Identify the bounded region

The bounded region is identified as the finite area enclosed between the bell-shaped curve, the vertical lines and the x-axis.

Step 3: Establish integral boundaries

The desired area can be obtained by integrating the function \(y=x e^{-x^{2}}\) from x=0 (starting point) to x=1 (ending point).

Step 4: Evaluate the definite integral

The area is determined by solving the definite integral \( \int_{0}^{1} x e^{-x^{2}} dx \). You have to apply integration by substitution here, let \( u = x^2 \) then \( du = 2x dx\). After expressing everything in terms of u, evaluate the integral.

Step 5: Simplify the evaluated integral

Upon simplification \( \int_{0}^{1} x e^{-x^{2}} dx = [-0.5*e^{-x^{2}}]_{0}^{1} = -0.5e^{-1} - (-0.5) = 0.5 - 0.5/e \)

Key Concepts

Definite Integral

A **definite integral** is an essential concept in calculus that helps us find the area under a curve on a particular interval. When you think of a definite integral, visualize it as a "tool" or "technique" used to calculate the exact accumulation of quantities, like area. In the exercise, the definite integral \[ \int_{0}^{1} x e^{-x^{2}} \, dx \]is used to find the area under the curve of the function \( y=x e^{-x^{2}} \) from \( x=0 \) to \( x=1 \).

• The limits \( 0 \) and \( 1 \) are called the lower and upper limits, respectively.
• The integral sign \( \int \) indicates the process of integration.
• The function \( x e^{-x^{2}} \) is referred to as the integrand.

Evaluating this integral results in the value that represents the area between the curve and the x-axis over the specified interval.This process involves adding up infinitely many infinitesimally small areas under the curve between the two limits. Each small area is represented by "tiny rectangles", and the sum of these rectangles gives the total area under the curve.

Integration by Substitution

**Integration by Substitution** is a method used to simplify the process of integration. It is like the reverse of the chain rule in differentiation. This technique is particularly useful when the integrand is a product of functions that fits the form of a known derivative. In the given exercise, you need to evaluate \[ \int x e^{-x^{2}} \, dx \] by making a strategic substitution.Here are the steps to follow:

• Identify a part of the integrand that can be substituted with a single variable. In this exercise, let \( u = x^2 \).
• Calculate \( du \). Since \( du = 2x \, dx \), express this as \( x \, dx = \frac{1}{2} \, du \).
• Replace all \( x \) terms in the integral, giving you \( \int e^{-u} \times \frac{1}{2} \, du \).

Once the substitution is complete, integrate with respect to \( u \). After evaluating the integral, substitute back the original variables if necessary.

Area Under a Curve

The concept of finding the **area under a curve** is very significant in calculus, especially in understanding and applications related to accumulation. The area represents a quantitative measure of the "space" beneath the curve on a graph. In our exercise, we're interested in determining the area that's bounded by

• The function \( y = x e^{-x^{2}} \)
• The lines \( x = 0 \) and \( x = 1 \)
• The x-axis (\( y = 0 \))

This is equivalent to calculating the definite integral \( \int_{0}^{1} x e^{-x^{2}} \, dx \). This integral gives the exact size of the area between the specified bounds. Finding areas in this way has applications in physics, engineering, and probability, where you need to know how a quantity accumulates over an interval of time or space. Visualizing the area helps make sense of how functions behave and how changes happen within fixed boundaries.