Problem 39 Use a definite integral to find ... [FREE SOLUTION]

Chapter 5: Problem 39

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

Short Answer

Expert verified

The area is \(2(e - 1)\).

Step by step solution

Setup the Definite Integral

To find the area under the curve of the function \(f(x) = e^{x/2}\) from \(x = 0\) to \(x = 2\), we express this as a definite integral: \[ \int_{0}^{2} e^{x/2} \, dx \] This integral will calculate the total area under the curve between the specified \(x\)-values.

Integrate the Function

Determine the antiderivative of \(e^{x/2}\). The antiderivative is found by considering the substitution \(u = \frac{x}{2}\), which gives \(du = \frac{1}{2}dx\) or \(dx = 2du\). Substituting, we have \[ \int e^{x/2} \, dx = 2 \int e^u \, du = 2e^u + C = 2e^{x/2} + C \] This is the antiderivative of \(e^{x/2}\) with respect to \(x\).

Apply the Definite Integral

Now, apply the limits of integration from \(0\) to \(2\):\[ \int_{0}^{2} e^{x/2} \, dx = \left[ 2e^{x/2} \right]_{0}^{2} \] Substitute the upper limit and the lower limit into the antiderivative and find the difference.

Calculate the Area

Substitute \(x = 2\) into the antiderivative: \[ 2e^{2/2} = 2e^{1} = 2e \]Substitute \(x = 0\) into the antiderivative: \[ 2e^{0/2} = 2e^{0} = 2 \times 1 = 2 \] Now find the area by calculating the difference: \[ 2e - 2 \] This simplifies to \( 2(e - 1) \).

Sketch the Curve

To sketch the function, plot \(f(x) = e^{x/2}\) on a coordinate grid for the range \(x = 0\) to \(x = 2\). The curve starts at \(e^0 = 1\) at \(x = 0\) and increases gradually to \(e\) at \(x = 2\). Shade the area under the curve from \(x = 0\) to \(x = 2\) to represent the region whose area we computed.

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Key Concepts

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

Area under a curve

When you talk about finding the area under a curve, you are essentially trying to calculate the space between the curve of a function and the x-axis within specified limits. This is useful in various applications such as determining the total accumulation of a quantity at a given point, like distance, energy, or mass.

Let鈥檚 break down how you can calculate this area for the function given: \( f(x) = e^{x/2} \) between \( x = 0 \) and \( x = 2 \).

Firstly, you express this calculation as a definite integral: \( \int_{0}^{2} e^{x/2} \, dx \).
The definite integral takes care of summing up all the tiny slices or sections of area under the curve across the specified range.
This sum provides the exact area under the curve, within the limits.

By using definite integrals, you obtain a very accurate measurement of this area, which is essential in mathematics and physics, where precise calculations are crucial.

Antiderivatives

An antiderivative, also known as an indefinite integral, is essentially the reverse of taking a derivative. When you find an antiderivative, you are looking for a function whose derivative will give you the original function.

In the context of the given problem, you needed to find the antiderivative of \( e^{x/2} \) to solve the definite integral.

To track back to the antiderivative of \( e^{x/2} \), a substitution method is often utilized.
You use a substitution such as \( u = \frac{x}{2} \) which simplifies the integration process.
This leads you to a more basic integral with respect to \( u \), namely \( 2 \int e^u \, du \).

Finding the antiderivative in this way ensures that you can transform any difficult integral into simpler tasks, harnessing the power of reverse differentiation.

Exponential Functions

Exponential functions are a crucial part of calculus and mathematical analysis, representing continuous growth or decay. These functions have the form \( f(x) = a^{g(x)} \), where a is a positive constant. In our example, \( f(x) = e^{x/2} \), the base of the exponential function is \( e \), the exponential constant approximately equal to 2.71828.

Exponential functions pose some interesting properties:

They are continuously increasing (if the exponent is positive) or decreasing (if the exponent is negative).
They are used extensively to model real-life phenomena like population growth, radioactive decay, and even in financial calculations for compound interest.
In the function \( e^{x/2} \), the growth rate depends on the variable \( \frac{x}{2} \), modifying how quickly or slowly the function grows.

Understanding exponential functions richens your grasp of many processes in nature and technology, helping assess evolving systems succinctly. These functions build the foundation of many advanced topics in calculus and beyond!

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

Short Answer

Step by step solution

Setup the Definite Integral

Integrate the Function

Apply the Definite Integral

Calculate the Area

Sketch the Curve

Key Concepts

Area under a curve

Antiderivatives

Exponential Functions

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