Problem 10 - Calculus and its applications

Question

Find the area under the given curve over the indicated interval. $$ y=e^{x} ; \quad[0,2] $$

Step-by-Step Solution

Answer

The area under the curve is \( e^2 - 1 \).

Step 1: Identify the Function

The given problem asks us to find the area under the curve for the function \( y = e^x \) over the interval \([0, 2]\). This means we need to calculate the definite integral of \( e^x \) from 0 to 2.

Step 2: Set Up the Definite Integral

The area under the curve \( y = e^x \) from \( x=0 \) to \( x=2 \) is found by evaluating the definite integral \( \int_{0}^{2} e^x \, dx \).

Step 3: Integrate the Function

To find the integral of \( e^x \), we use the basic integral formula \( \int e^x \, dx = e^x + C \), where \( C \) is the constant of integration. For a definite integral, we do not use the constant. Thus, we calculate \( F(x) = e^x \).

Step 4: Evaluate the Definite Integral

Apply the limits of the interval to the function \( F(x) = e^x \). Evaluate \( F(b) - F(a) \), where \( a = 0 \) and \( b = 2 \). This becomes \( F(2) - F(0) = e^2 - e^0 \).

Step 5: Simplify the Expression

Simplify the expression \( e^2 - e^0 \). Since \( e^0 = 1 \), the expression simplifies to \( e^2 - 1 \). This is the exact area under the curve from 0 to 2.

Key Concepts

Area under the curve

When we talk about finding the "area under the curve," we are talking about determining the space that lies between a curve and the x-axis over a certain interval. This concept is very important in calculus, particularly when dealing with functions and their graphs. What we are essentially doing is calculating how much space is enclosed by the curve between two points on the x-axis. In the case of exponential functions like \( y = e^x \), this involves finding the integral over a specified range.The definite integral is the tool we use to figure out this area. For a given function \( f(x) \) defined over an interval \([a, b]\), the definite integral \( \int_{a}^{b} f(x) \; dx \) computes the total area under \( f(x) \) from \( x = a \) to \( x = b \).

• The definite integral considers both the curve and the x-axis.
• If the curve is above the x-axis, the area is considered positive.
• If below, it is negative, but in area calculation contexts, we treat it consistently to represent physical space.

By using this method, the calculated value comes with the units of the graph's axes, such as square units if the axes have the same units.For the exponential function \( y = e^x \), the area under the curve from 0 to 2 is calculated to be \( e^2 - 1 \). This states that the total space between the curve and the x-axis, over this interval, amounts to \( e^2 - 1 \) units.

Exponential functions

Exponential functions, especially fundamental in calculus and real-world applications, are functions of the form \( y = a \cdot e^{kx} \) where \( e \) is approximately 2.71828, known as Euler's number. These functions increase or decrease at rates proportional to their current values. In simpler terms, exponential functions can grow very fast, which makes them essential in modeling population growth, radioactive decay, and interest calculations in finance.In the given exercise, the function \( y = e^x \) is one of the simplest forms of exponential functions. Here:

• "\( a \)" is 1, meaning there is no vertical stretching or compression.
• "\( k \)" is also 1, indicating a direct proportional increase in \( y \) as \( x \) increases.

Noticeably, this function passes through the point (0,1) since \( e^0 = 1 \). For each increasing value of \( x \), \( y \) becomes \( e^x \), which further highlights the escalating nature of exponential growth.Understanding exponential functions provides valuable insight into how swiftly things can grow geometrically, as opposed to linearly. These characteristics explain why, as \( x \) increases from 0 to 2, the function \( e^x \) varies rapidly, influencing a larger area beneath the curve.

Integral Calculus

Integral calculus is a major branch of calculus that deals with the concept of integration. Integration is about assimilating parts to form a whole sum, which in mathematical terms, involves calculating the area under a curve or finding the anti-derivative of a function.There are two main types of integrals in calculus:

• Indefinite Integral: Represents a family of functions and includes an arbitrary constant \( C \). It's written as \( \int f(x) \, dx \).
• Definite Integral: Provides a numerical value representing the area under the curve between two specific points. It's denoted \( \int_{a}^{b} f(x) \, dx \).

In this exercise, to find the area under the curve of \( y = e^x \) from 0 to 2, we work with the definite integral \( \int_{0}^{2} e^x \, dx \).Evaluating these involves two key processes:

• Finding the integral (like \( e^x \)) as the anti-derivative of the function.
• Applying the limits of the definite integral, and computing the difference between these values to get \( F(b) - F(a) \), which here becomes \( e^2 - e^0 \).

Integral calculus plays a crucial role not just in theoretical mathematics, but also in fields like physics, engineering, and statistics, offering ways to tackle calculations involving accumulated change or area.