Problem 68 - Applied Calculus

Question

Find the area between the curve \(y=x e^{x^{2}}\) and the \(x\) -axis from \(x=1\) to \(x=3\) (Leave the answer in its exact form.)

Step-by-Step Solution

Answer

The exact area is \( \frac{1}{2} ( e^{9} - e ) \).

Step 1: Understand the Problem

We are tasked with finding the area between the curve defined by the function \( y = x e^{x^2} \) and the \( x \)-axis from \( x = 1 \) to \( x = 3 \). This area can be found using definite integration.

Step 2: Set Up the Integral

To find the area under the curve from \( x = 1 \) to \( x = 3 \), we need to evaluate the definite integral \( \int_{1}^{3} x e^{x^2} \, dx \).

Step 3: Use Substitution

Let \( u = x^2 \), so that \( du = 2x \, dx \) or \( x \, dx = \frac{1}{2} \, du \). The limits also change from \( x=1 \) to \( u=1^2=1 \), and from \( x=3 \) to \( u=3^2=9 \).

Step 4: Substitute Variables and Integrate

Rewrite the integral in terms of \( u \): \[ \int_{1}^{9} e^{u} \frac{1}{2} \, du = \frac{1}{2} \int_{1}^{9} e^{u} \, du \]Integrate to get \( \frac{1}{2} ( e^{u} ) \Bigg|_{1}^{9} \).

Step 5: Evaluate the Integral

Evaluate the definite integral:\[ \frac{1}{2} ( e^{9} - e^{1} ) \]Which simplifies to \( \frac{1}{2} ( e^{9} - e ) \). This is the exact area between the curve and the \( x \)-axis.

Key Concepts

Area Under a Curve

When we talk about the "area under a curve," we're referring to the space between a curve described by a function and the x-axis. This area represents the integral of that function over a specified interval. In mathematical terms, the area is found using definite integration. Given a function \( y = f(x) \), the area under the curve from \( x = a \) to \( x = b \) is given by the definite integral \( \int_{a}^{b} f(x) \, dx \).
For the problem at hand, the function is \( y = x e^{x^2} \), and we're interested in the interval from \( x = 1 \) to \( x = 3 \). Finding this area involves setting up and evaluating the definite integral \( \int_{1}^{3} x e^{x^2} \, dx \). This integral calculates the total area between the curve and the x-axis over that interval.

Integration by Substitution

Integration by substitution is a method used to simplify integrals by making a substitution that transforms the integral into a simpler form. It's analogous to the reverse of the chain rule used in differentiation.
For our integral \( \int_{1}^{3} x e^{x^2} \, dx \), we use the substitution technique to handle the complex expression \( e^{x^2} \). By setting \( u = x^2 \), we find that \( du = 2x \, dx \), or rearrange it to find \( x \, dx = \frac{1}{2} \, du \). This substitution simplifies the integration process.
Subsequently, the limits of integration must also change according to our substitution. When \( x = 1 \), \( u = 1^2 = 1 \), and when \( x = 3 \), \( u = 3^2 = 9 \). Thus, the integral becomes \( \frac{1}{2} \int_{1}^{9} e^{u} \, du \), a simpler form that can be integrated directly.

Exponential Function

The exponential function, denoted as \( e^x \), is a mathematical function that grows rapidly with increasing \( x \). It is unique because its rate of growth is proportional to its current value. Integration involving an exponential function often appears in calculus problems, like the one we're solving.
For the integration problem, the function \( y = x e^{x^2} \) contains the exponential term \( e^{x^2} \). Through substitution, we rewrote this in terms of \( u \) as \( e^u \). The integral of \( e^u \) with respect to \( u \) is straightforward: \( \int e^u \, du = e^u + C \). In a definite integral, we simply evaluate the antiderivative from the lower to the upper limit of \( u \). Hence, solving \( \frac{1}{2} ( e^{u} ) \Bigg|_{1}^{9} \) results in \( \frac{1}{2} ( e^{9} - e^{1} ) \), yielding the exact area.