Problem 53 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Set Up the Integral

To find the area between the curve \( y = e^{-ax} \) and the \( x \)-axis from \( x = 0 \) to \( \infty \), you need to set up a definite integral. The formula for the area \( A \) is given by:\[\int_{0}^{\infty} e^{-ax} \, dx\]

Step 2: Find the Antiderivative

The next step is to find the antiderivative of the function \( e^{-ax} \). The antiderivative of \( e^{-ax} \) is \[ -\frac{1}{a} e^{-ax} + C \] where \( C \) is the constant of integration.

Step 3: Evaluate the Definite Integral

Apply the limits of integration from \( 0 \) to \( \infty \): \[ A = \left. -\frac{1}{a} e^{-ax} \right|_{0}^{\infty} \]

Step 4: Evaluate at Infinity and Zero

First, evaluate \( \lim_{x \to \infty} -\frac{1}{a} e^{-ax} = 0 \), since \( e^{-ax} \) approaches \( 0 \) as \( x \) goes to infinity. Then evaluate \( -\frac{1}{a} e^{-ax} \) at \( x = 0 \), which gives \( -\frac{1}{a} \cdot e^{0} = -\frac{1}{a} \).

Step 5: Compute the Area

Now, subtract the values found in Step 4: \[ A = 0 - \left(-\frac{1}{a}\right) = \frac{1}{a} \] This is the area between the curve and the \( x \)-axis from \( x = 0 \) to \( \infty \).

Key Concepts

Antiderivatives

In integral calculus, an antiderivative, also known as an indefinite integral, is essentially the reverse of differentiation. It is a function whose derivative equals the original function we started with.
Finding the antiderivative allows us to determine the area under a curve or solve differential equations.
When dealing with an exponential function such as \( y = e^{-ax} \), the process of finding the antiderivative involves determining a function that will differentiate back to \( e^{-ax} \).To find the antiderivative of \( e^{-ax} \), we look for a function whose derivative is equal to \( e^{-ax} \). This leads us to:

• The antiderivative of \( e^{-ax} \) is \( -\frac{1}{a} e^{-ax} \), where \( a \) is a constant and \( a > 0 \).

Understanding this concept is crucial because it provides the foundation for calculating definite integrals, which are used to find areas under curves, or to evaluate the total amount of a quantity over an interval.

Definite Integral

The definite integral represents the accumulation of quantities, such as area, volume, or other measures over an interval. It is expressed as: \[ \int_{a}^{b} f(x) \, dx \] where \( a \) and \( b \) are the limits of integration that define the interval.In our specific exercise for the function \( y = e^{-ax} \), we determine the area under the curve from \( x = 0 \) to \( x = \infty \):

• We set up the definite integral \( \int_{0}^{\infty} e^{-ax} \, dx \).

Evaluating a definite integral involves calculating the antiderivative of the function and then applying the Fundamental Theorem of Calculus. This theorem helps derive the total area by evaluating the antiderivative at the boundaries of the interval. By computing \( \lim_{x \to \infty} -\frac{1}{a} e^{-ax} \) and noting it approaches zero, and \( -\frac{1}{a} e^{-ax} \) at \( x = 0 \), which simplifies to \(-\frac{1}{a}\), we can subtract these two results to find the definite integral value, which is \( \frac{1}{a} \).

Area Under a Curve

Calculating the area under a curve is a major application of definite integrals. It allows us to determine how much accumulated quantity, such as area, exists between two limits on a graph.
The concept is particularly useful in physics, engineering, and economics, where such accumulations can represent everything from total distance traveled to cost analysis.For the curve given by \( y=e^{-ax} \), the area between this curve and the x-axis from \( x = 0 \) to \( x = \infty \) is computed using integration.
Here, the integral \( \int_{0}^{\infty} e^{-ax} \, dx \) maps the accumulated area as \( x \) approaches infinity.

• Evaluating it shows that the entire area equals \( \frac{1}{a} \), illustrating the finite area occupied even as the curve stretches to infinity.

This serves to highlight how exponential decay functions can level off, creating a finite area under what appears to be an infinitely extending graph.

Exponential Functions

Exponential functions are a key type of mathematical function with a constant base raised to a variable exponent, expressed generally as \( y = b^x \). However, when the exponent involves a negative variable, such as in \( y = e^{-ax} \) with \( a > 0 \), it represents exponential decay.This type of function is characterized by:

• A rapid decrease in function value as \( x \) increases.
• Approaches zero but never quite reaches it, a property that is pivotal in computing limits and integrals.

The properties of these functions make them extensively applicable, modeling a wide range of real-life phenomena such as radioactive decay, cooling processes, and population dynamics.
In our exercise, the exponential decay is essential in setting the scene for the integral, as it ensures that the area under the curve between 0 and infinity is finite, leading to the solution \( \frac{1}{a} \). This demonstrates not only the mathematical properties but also the practical utility of exponential functions in understanding real-world behavior.