Problem 64 - Applied Calculus

Question

GENERAL: Internet Host Computers The number of Internet host computers (computers connected directly to the Internet, for networks, bulletin boards, or online services) has been growing at the rate of \(f(x)=x e^{0.1 x}\) million per year, where \(x\) is the number of years since \(1990 .\) Find the total number of Internet host computers that will have been added from 1990 to \(2020 .\)

Step-by-Step Solution

Answer

Approximately 4117 million Internet host computers were added from 1990 to 2020.

Step 1: Define the function and interval

The growth rate of Internet host computers is given by the function \( f(x) = x e^{0.1x} \), where \( x \) is the number of years since 1990. We need to find the total number added between 1990 and 2020. So, the interval for \( x \) is from 0 to 30 (since 2020 corresponds to \( x = 30 \)).

Step 2: Set up the integral for total number added

To find the total number of Internet host computers added, we need to calculate the definite integral of the function \( f(x) \) from \( x = 0 \) to \( x = 30 \). The integral is: \[ \int_{0}^{30} x e^{0.1x} \, dx \]

Step 3: Integrate the function

To integrate \( x e^{0.1x} \), use integration by parts: let \( u = x \) and \( dv = e^{0.1x} \, dx \). Then, \( du = dx \) and \( v = \frac{1}{0.1} e^{0.1x} = 10 e^{0.1x} \).

Step 4: Apply integration by parts formula

Using integration by parts, \( \int u \, dv = uv - \int v \, du \), we get: \[ \int x e^{0.1x} \, dx = x \cdot 10 e^{0.1x} - \int 10 e^{0.1x} \, dx \] Computing the remaining integral: \[ \int 10 e^{0.1x} \, dx = 10 \cdot 10 e^{0.1x} = 100 e^{0.1x} \] Hence, \[ \int x e^{0.1x} \, dx = 10x e^{0.1x} - 100 e^{0.1x} + C \]

Step 5: Evaluate the definite integral

Now, evaluate the expression at the bounds 30 and 0: \[ \left[ 10x e^{0.1x} - 100 e^{0.1x} \right]_{0}^{30} \]At \( x = 30 \):\( 10 \cdot 30 e^{3} - 100 e^{3} = 300 e^{3} - 100 e^{3} = 200 e^{3} \)At \( x = 0 \):\( 10 \cdot 0 e^{0} - 100 e^{0} = -100 \)Thus, the total number is:\[ 200 e^{3} + 100 \]

Step 6: Calculate the numerical value

Compute the exponential term \( e^{3} \) using a calculator and complete the computation: \( e^{3} \approx 20.08554 \), hence \( 200 \cdot 20.08554 + 100 = 4017.11 + 100 \approx 4117.11 \). So, approximately 4117 million Internet host computers were added from 1990 to 2020.

Key Concepts

Definite Integral

The definite integral is a powerful mathematical tool used to calculate the accumulation of quantities. When dealing with functions that describe rates of change, such as the growth rate of Internet host computers, the definite integral helps us find the total change over a specific interval. In our case, the growth rate function is given by \( f(x) = x e^{0.1x} \), representing the millions of Internet host computers added per year since 1990.

To find the total number of computers added between 1990 and 2020, we set up the definite integral of \( f(x) \) from \( x = 0 \) to \( x = 30 \). Mathematically, this is expressed as:

• \[ \int_{0}^{30} x e^{0.1x} \, dx \]

The definite integral not only helps us quantify the total growth but also provides insights into how the rate of growth behaves over time. In computational calculus, calculating such integrals often involves advanced techniques, especially when functions are complex.

Computational Calculus

Computational calculus refers to the application of numerical methods and algorithms to solve calculus problems, especially when dealing with complex integrals that do not have straightforward analytical solutions. For the exercise on Internet growth, we employ integration by parts to tackle the integral where the function consists of a product of two simpler functions, \( x \) and \( e^{0.1x} \).

The technique of integration by parts is used here, which stems from the product rule for differentiation. In this context, it allows us to break down the integral into manageable pieces. We choose:

• \( u = x \)
• \( dv = e^{0.1x} \, dx \)

Thus, \( du = dx \) and \( v = 10 e^{0.1x} \). The integration by parts formula, \( \int u \, dv = uv - \int v \, du \), is applied to obtain:

• \[ \int x e^{0.1x} \, dx = 10x e^{0.1x} - \int 10 e^{0.1x} \, dx \]

This leads us to another integral that yields a simple solution: \( \int 10 e^{0.1x} \, dx = 100 e^{0.1x} \). Completing the integration provides a function that we then evaluate over the provided bounds of the definite integral.

Internet Growth Rate

Understanding the concept of the internet growth rate is essential when analyzing how quickly technology and infrastructure evolve. In this exercise, the growth rate of Internet host computers—which can be interpreted as the speed at which new computers are added each year—is modeled by the function \( f(x) = x e^{0.1x} \). This function incorporates exponential growth, reflecting an increasing rate over time.

Exponential growth is characteristic of technological advancements and networks. It indicates that as time progresses, not only do more computers get connected, but the number of connections grows at an accelerating pace. From 1990 to 2020, this model helps us predict that the growth is not linear—earlier years see fewer additions compared to later years.

Analyzing this function emphasizes the importance of exponential terms when predicting future trends. By calculating the definite integral of this growth function from 0 to 30 (years since 1990), we estimate a total of approximately 4117 million new host computers. This substantial increase highlights the rapid pace of internet expansion during these three decades.