Problem 37 Use a definite integral to find ... [FREE SOLUTION]

91影视

Brief Applied Calculus

Geoffrey C. Berresford, Andrew M. Rockett

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 5: Problem 37

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. $f(x)=2 e^{x}$ from $x=0$ to $x=\ln 3$

Short Answer

Expert verified

The area is 4.

Step by step solution

Understand the problem

We need to find the area under the curve of the function $ f(x) = 2e^x $ between $ x = 0 $ and $ x = \ln 3 $. We will use a definite integral to find this area.

Set up the definite integral

We set up the integral for the function $ f(x) = 2e^x $ from $ x = 0 $ to $ x = \ln 3 $: \[ \int_{0}^{\ln 3} 2e^x \, dx \]

Integrate the function

To integrate $ 2e^x $, recall that the integral of $ e^x $ with respect to $ x $ is $ e^x $. Therefore, the integral of $ 2e^x $ is:\[ \int 2e^x \, dx = 2e^x + C \] where $ C $ is the constant of integration, which we do not need for definite integrals.

Evaluate the definite integral

Now substitute the limits $ x = \ln 3 $ and $ x = 0 $ into the integrated function: \[2e^{\ln 3} - 2e^0 = 2 \times 3 - 2 \times 1 = 6 - 2 = 4\]

Sketch the curve and the region

The curve of $ f(x) = 2e^x $ is an exponential curve that starts at (0,2) when $ x = 0 $. At $ x = \ln 3 $, the function value is 6, so the curve passes through ($ \ln 3 $, 6). The region under the curve from $ x = 0 $ to $ x = \ln 3 $ forms a bounded area which is computed to be 4. Sketch this curve and shade the region between these x-values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Area Under Curve

Understanding the area under a curve is a key part of integral calculus. When you're working with a function like $ f(x) = 2e^x $, finding the area under its curve on a specific interval tells you about the accumulation of values over that range.

This can be interpreted in many real-world applications, such as calculating total revenue over time or finding the distance traveled. To find this area, we use a definite integral, which involves calculating the integral of the function across a certain interval of $x$.

First, you label the function and the interval. For our exercise, the interval is from $x = 0$ to $x = \ln 3$.
Next, set up the definite integral, $ \int_{0}^{\ln 3} 2e^x \, dx $, to represent the area you want to find.
Finally, solve this integral and evaluate it using the limits provided. This process gives the area under the curve between the specified $x$-values, which in this case is 4 square units.

Exponential Functions

Exponential functions, such as $ f(x) = e^x $, grow at an increasing rate and are shown as curved lines on a graph. In our exercise, we have an exponential function with a multiplier, $ 2e^x $.

This multiplier stretches the graph vertically, making it rise more sharply as $ x $ increases.

For $ 2e^x $, at $ x = 0 $, the value begins at 2, since anything raised to the power of 0 is 1, making it $2 \times 1 = 2$.
Exponential functions are powerful in modeling growth processes, including populations, radioactive decay, and compound interest due to their rapid increase.
Graphically, this curve is smooth, continuous, and increases without bounds as $ x $ grows larger.

By integrating an exponential function, we can find accumulated values of these processes over a certain interval, illustrating their importance in calculus.

Integral Calculus

Integral calculus is a branch of mathematics focused on the concept of accumulation. It often helps in finding areas, volumes, and other quantities under curves. In particular, definite integrals provide a precise method to calculate these quantities for bounded curves.

With our exercise, integral calculus allows us to find the total area under $ f(x) = 2e^x $ from $ x = 0 $ to $ x = \ln 3 $.

The definite integral $ \int_{a}^{b} f(x) \, dx $ represents the accumulated area from $ x = a $ to $ x = b $, using the integrated function value.
The definite integral has two parts: finding the anti-derivative and applying the limits to this anti-derivative function.
In practical terms, it translates the concept of a curve's area into precise numerical values, offering solutions to real-world problems involving density, force, and other accumulated quantities.

Integral calculus is a fundamental tool in mathematics that expands our ability to compute and interpret data across various scientific fields.

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91影视

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. \(f(x)=2 e^{x}\) from \(x=0\) to \(x=\ln 3\)

Short Answer

Step by step solution

Understand the problem

Set up the definite integral

Integrate the function

Evaluate the definite integral

Sketch the curve and the region

Key Concepts

Area Under Curve

Exponential Functions

Integral Calculus

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