Problem 24 - 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 under the curve is \( e - 1 \).

Step 1: Understand the Function and Interval

We are given the function \( f(x) = e^x \) and need to find the area under this curve from \( x = 0 \) to \( x = 1 \). This involves computing the definite integral of \( f(x) \) over the interval \([0, 1]\).

Step 2: Set Up the Definite Integral

The area under the curve \( f(x) = e^x \) from \( x = 0 \) to \( x = 1 \) is given by the definite integral:\[\int_{0}^{1} e^x \, dx\]

Step 3: Integrate the Function

To integrate \( e^x \), recall that the antiderivative of \( e^x \) is \( e^x \). Thus, the indefinite integral of \( e^x \) is:\[\int e^x \, dx = e^x + C\]Where \( C \) is the constant of integration.

Step 4: Evaluate the Definite Integral

Now we evaluate the integral from \( x = 0 \) to \( x = 1 \):\[\int_{0}^{1} e^x \, dx = \left[ e^x \right]_{0}^{1} = e^1 - e^0\]Since \( e^0 = 1 \) and \( e^1 = e \), this simplifies to:\[e - 1\]

Step 5: Provide the Interpretation

The value \( e - 1 \) represents the area under the curve \( y = e^x \) from \( x = 0 \) to \( x = 1 \). This result is the actual area within the region bounded by the curve, the x-axis, and the vertical lines \( x = 0 \) and \( x = 1 \).

Step 6: Visualize with a Sketch

To visualize, sketch the curve \( y = e^x \) which is increasing exponentially. Plot the points at \( x = 0 \) where \( y = 1 \) and \( x = 1 \) where \( y = e \). Shade the area between the curve, the x-axis, and the vertical lines \( x = 0 \) and \( x = 1 \).

Key Concepts

Understanding the Area Under a Curve

Calculating the area under a curve is a fundamental concept in calculus. This area can be described as the region enclosed between a curve, the x-axis, and vertical lines at the endpoints of an interval.

• This region represents a definite integral of the function over the specified interval.
• The area provides us with valuable geometric interpretation and practical application, such as finding the total quantities when rates are involved.

In the context of this exercise, finding the area under the curve of the exponential function from \(x = 0\) to \(x = 1\) involves integration. This requires understanding both the function's behavior and the limits of integration.

The Exponential Function Explained

The exponential function, commonly denoted as \(f(x) = e^x\), is one of the most important functions in mathematics. Its base, \(e\), is an irrational number approximately equal to 2.71828.

• It models phenomena with constant growth rate, such as population dynamics or radioactive decay.
• Characteristic of this function is its increasing rate of change, which implies the slope becomes steeper as \( x \) increases.
• This function always yields positive values when \( x \) is real.

In the problem, the exponential nature manifests in the curve showing sharp increase starting from 1 at \(x=0\) up to \(e\) at \(x=1\).

Integral Evaluation for Finding Area

Evaluating a definite integral involves computing the antiderivative and then applying the Fundamental Theorem of Calculus.

• For \( f(x) = e^x\), the antiderivative is the function itself, \( \int e^x \, dx = e^x + C \), where \( C \) is a constant.
• Definite integration, however, negates the constant \( C \) since it evaluates the difference over an interval.

For our exercise, we compute: \[ \int_{0}^{1} e^x \, dx = \left[ e^x \right]_{0}^{1} = e^1 - e^0 = e - 1 \] Here, \( e^0 = 1 \), which simplifies the result. The outcome \( e - 1 \) is the precise area under the curve within the specified bounds.

The Importance of Function Sketching

Sketching a function is crucial to visually interpret and understand the behavior of a graph. For exponential functions like \( y = e^x \), mapping a graph ensures comprehension of how they change over intervals.

• Start by identifying key points: at \( x = 0 \), \( y = 1 \) and at \( x = 1 \), \( y = e \).
• The curve will rise sharply through these points due to the exponential nature.
• Use this sketch to shade the area between the curve, the x-axis, and vertical lines \( x = 0 \) and \( x = 1 \) to visualize the region we calculated.

This exercise aids in correlating the algebraic integration process with its graphical representation, reinforcing understanding and retention of concepts.