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Chapter 13: Problem 8

Compute the following definite integrals: \(\int_{1}^{4} \frac{4}{\sqrt{x}} d x\)

Short Answer

Expert verified
The definite integral of the given function from 1 to 4 is 8.

Step by step solution

01

Identify the function to integrate

Here, the function to integrate is \(f(x) = \frac{4}{\sqrt{x}}\). Notice this can alternatively be written as \(f(x) = 4x^{-1/2}\), where the power rule for integrals can be applied.
02

Carry out the integration

Apply the power rule for integration, which states that the integral of \(x^n\) with respect to \(x\) is \(\frac{1}{n+1} x^{n+1}\) (where \(n \neq -1\)). Here, \(n = -1/2\), so the integral becomes \(\frac{4}{-1/2 + 1} x^{-1/2 + 1} = 8x^{1/2}\).
03

Apply the limits of integration

Now, the limits of integration from 1 to 4 have to be applied. This means to substitute these into the integrated function and calculate the difference: \[ 8 \cdot 4^{1/2} - 8 \cdot 1^{1/2} = 8 \cdot 2 - 8 \cdot 1 = 16 - 8 = 8 \]

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Key Concepts

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

Power Rule for Integrals
When tackling integrals, one of the simplest and most powerful tools available is the power rule. It allows us to integrate functions of the form \(x^n\) with relative ease.
For a function \(x^n\), its integral is given by:
  • Formula: \( \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \)
Here, \(n\) should not be \(-1\) since that would require a different method involving the natural logarithm function. In the original problem, the expression \(\frac{4}{\sqrt{x}}\) can be rewritten using exponents as \(4x^{-1/2}\). Thus, you apply the power rule:
  • Calculation: \( \frac{4}{-1/2 + 1} x^{-1/2 + 1} = 8x^{1/2} \)
This results in a new exponents that makes the integration process simple and clear.
Limits of Integration
Once an integral is calculated, applying the limits of integration is crucial. These limits define the interval over which the area under the curve is calculated.
In definite integrals like the one in this exercise, the limits of integration are specified at the bottom and top of the integral symbol. Here's how it works:
  • Lower Limit: The bottom number \(1\) here represents the starting point of integration on the x-axis.
  • Upper Limit: The top number \(4\) signifies where to stop integrating on the x-axis.
To apply these limits, substitute them into the integrated function, compute those values, and find their difference:
  • Computation: \(8 \cdot 4^{1/2} - 8 \cdot 1^{1/2} = 16 - 8 = 8\)
This gives the total area and is the final result of the definite integral.
Integration Techniques
Integrating functions involves multiple techniques and strategies. The goal is to simplify complex expressions and facilitate the integration process.
Here are some common techniques that prove beneficial:
  • Rewriting Functions: Changing the form of the function, as seen with \(\frac{4}{\sqrt{x}}\) to \(4x^{-1/2}\), which makes applying the power rule straightforward.
  • Substitution: Not applicable in this particular problem, but often essential for handling more complicated functions.
  • Integration by Parts: Utilized when dealing with products of functions where simpler methods are unsuitable.
Understanding and mastering these techniques is invaluable for efficiently solving integration problems, whether they're definite or indefinite integrals.

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