Problem 38 - Applied Calculus

Question

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=e^{-x} \text { from } x=0 \text { to } x=1 $$

Step-by-Step Solution

Answer

The area is \(\frac{e-1}{e}\).

Step 1: Set Up the Definite Integral

The area under the curve of the function from \(x=a\) to \(x=b\) can be found using the definite integral \( \int_{a}^{b} f(x) \, dx \). Here, \( f(x) = e^{-x} \), \( a = 0 \), and \( b = 1 \). Thus, we need to compute: \[ A = \int_{0}^{1} e^{-x} \, dx \].

Step 2: Evaluate the Integral

The integral of \(e^{-x}\) with respect to \(x\) is \(-e^{-x}\). Thus, the definite integral becomes: \( \left[-e^{-x}\right]_{0}^{1} \).

Step 3: Apply the Limits of Integration

Substitute the upper limit \(x=1\) and the lower limit \(x=0\) into the antiderivative: \( -e^{-1} - (-e^{0}) \). This simplifies to \( -\frac{1}{e} + 1 \), since \( e^{0} = 1 \).

Step 4: Simplify the Expression

Subtract the terms: \( 1 - \frac{1}{e} = \frac{e}{e} - \frac{1}{e} = \frac{e - 1}{e} \).

Step 5: Sketch the Graph

Plot the graph of \(f(x) = e^{-x}\), which is a decreasing exponential function starting from 1 when \(x=0\) and approaching 0 as \(x\) increases. Shade the region under the curve between \(x=0\) and \(x=1\) to represent the area.

Key Concepts

Area Under the Curve

When we talk about finding the area under a curve in mathematics, we refer to the space between the curve itself and the x-axis on a graph, across a specified interval. This space can be calculated using definite integrals.

In the given exercise, our focus is on the function \(f(x)=e^{-x}\) from \(x=0\) to \(x=1\).
The definite integral helps us determine this exact area by accumulating tiny slices of area under the curve.

• Start by identifying the function, \(f(x)\).
• Locate the interval on the x-axis, from \(a\) to \(b\).
• Set up the definite integral \( \int_{a}^{b} f(x) \, dx \).

By solving this integral, you find the total area. It provides a way to quantify portions of graphs in terms of square units, an important concept in calculus.

Exponential Function

Exponential functions are a critical concept in mathematics. They have a distinct form: \(f(x) = a^x\), where a is a constant and the base of the exponent. In this exercise, the function \(f(x) = e^{-x}\) is an example where the base \(e\) is Euler's number, approximately 2.718.

Exponential functions can model several scenarios in real life, such as population growth or radioactive decay. In our function:

• The expression \(-x\) results in a decreasing curve.
• As \(x\) increases, \(f(x)\) approaches zero, creating a decaying curve.
• The curve is smooth and continuous, making it predictable over intervals.

Understanding these functions helps in analyzing their behavior under various x-values. For the integral calculation, knowing the antiderivative of \(e^{-x}\) is essential.

Limits of Integration

In calculus, the limits of integration specify the interval over which you calculate the area for a definite integral. These limits are two values that define where the integration starts and ends on the x-axis.

For our problem, these limits are:

• Lower limit \(a = 0\).
• Upper limit \(b = 1\).

They form the boundaries for our integration process, dictating the segment of the curve we're interested in. When we integrate \(f(x) = e^{-x}\) from \(0\) to \(1\), we only consider the behavior of the function between these two points.
After finding the antiderivative, we subtract its value at the lower limit from its value at the upper limit. This process is crucial for correctly solving the definite integral and obtaining the area. Understanding limits in integration ensures accurate computation of areas and is a fundamental part of integral calculus.