Problem 52 - Applied Calculus

Question

AREA Find the area between the curve \(y=e^{-4 x}\) and the \(x\) -axis from \(x=0\) to \(\infty\).

Step-by-Step Solution

Answer

The area is \(\frac{1}{4}\).

Step 1: Set up the integral

To find the area under the curve from \(x = 0\) to \(x = \infty\) for \(y = e^{-4x}\), we first set up the definite integral \( \int_{0}^{\infty} e^{-4x} \, dx \). This integral represents the area between the curve and the x-axis.

Step 2: Determine the antiderivative

Find the antiderivative of \(e^{-4x}\). The antiderivative of \(e^{ax}\) is \(\frac{1}{a} e^{ax}\), so here it becomes \(-\frac{1}{4} e^{-4x}\).

Step 3: Evaluate the improper integral

Since this is an improper integral extending to infinity, we evaluate it as a limit: \[ \lim_{b \to \infty} \int_{0}^{b} e^{-4x} \, dx = \lim_{b \to \infty} \left[ -\frac{1}{4} e^{-4x} \right]_{0}^{b} \].

Step 4: Apply the limits of integration

Substitute the limits into the antiderivative expression:\[ \lim_{b \to \infty} \left( -\frac{1}{4} e^{-4b} + \frac{1}{4} e^{0} \right) \].

Step 5: Simplify the expression

Simplify the expression by evaluating the limits:As \(b\) approaches infinity, \(e^{-4b}\) approaches zero. Therefore, the expression becomes:\[ 0 + \frac{1}{4}\].

Step 6: Determine the final area

This gives us the final area, which is \(\frac{1}{4}\).

Key Concepts

Definite Integrals

In calculus, definite integrals are used to find the area under a curve between two points, along the x-axis. The definite integral of a function from limit \(a\) to \(b\) is expressed as \(\int_{a}^{b} f(x) \, dx\). This tells us the net area between the curve of the function \(f(x)\) and the x-axis from \(x=a\) to \(x=b\). A definite integral gives a finite value for the area, capturing both positive and negative areas if the curve dips below the x-axis.

• The lower limit \(a\) and the upper limit \(b\) are the boundaries over which you integrate.
• If the function is continuous over \([a, b]\), the Fundamental Theorem of Calculus helps us find the integral.
• The result of a definite integral \(\int_{a}^{b} f(x) \, dx\) is the difference between the values of an antiderivative evaluated at \(b\) and \(a\).

When applying definite integrals in practical scenarios, we often find the areas of regions enclosed by curves, which is an essential skill in various fields such as physics, engineering, and economics.

Improper Integrals

Unlike definite integrals over a closed interval, improper integrals involve integration over an infinite interval or an unbounded function. They appear when we deal with functions stretching towards infinity, either continuously or at one endpoint (or both), as in our problem.

These integrals are handled using limits: substituting infinity with a variable like \(b\), performing the integration, and then taking the limit as \(b\) approaches infinity. For example, the integral \(\int_{0}^{\infty} e^{-4x} \, dx\) represents an improper integral due to the infinity limit. This approach lets us find areas under curves that extend indefinitely.

• Use limits to properly define the integral over an infinite range.
• The convergence of an improper integral may sometimes result in a finite area.
• If the limit does not exist, the improper integral is said to diverge to infinity.

The improper integrals are crucial in calculus because they allow access to calculating areas or probabilities throughout unbounded domains, essential in fields like probability theory and various branches of mathematics.

Exponential Functions

Exponential functions are a broad class of functions where the variable is in the exponent. The standard form of an exponential function is \(f(x) = a e^{bx}\), where \(e\) is Euler's number (approximately 2.718), and \(a\) and \(b\) are constants.

These functions model numerous growth and decay processes, making them valuable in science, economics, and engineering. In integration and differentiation, exponential functions have a unique characteristic that their derivatives and integrals are also exponential functions, scaled by their constants.

• The integral of an exponential function like \(e^{bx}\) is \(\frac{1}{b} e^{bx} + C\), where \(C\) is the integration constant.
• Exponential decay, like \(e^{-4x}\), rapidly decreases approaching zero, which is common in decay processes.
• These functions are continuous and differentiable, with properties making them ideal for modeling consistent, smooth transformations.

Understanding exponential functions' behavior, especially in terms of integration, is important in fields that require modeling sophisticated natural phenomena or financial calculations involving continuous compounding.