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Magazine Chapter 25: Problem 10
Evaluate the given definite integrals. $$\int_{1}^{5} 9 \sqrt{3 v+1} d v$$
Short Answer
Expert verified
The definite integral evaluates to 112.
Step by step solution
01
Recognize the Function and Bounds
The integral we need to evaluate is \( \int_{1}^{5} 9 \sqrt{3v + 1} \, dv \). This is a definite integral with bounds from 1 to 5.
02
Substitute to Simplify
To handle the square root, we'll use substitution. Let \( u = 3v + 1 \). Then, \( \frac{du}{dv} = 3 \), which implies \( dv = \frac{1}{3} du \). Our integral becomes \( \int 9 \sqrt{u} \cdot \frac{1}{3} \, du \), or \( 3 \int \sqrt{u} \, du \).
03
Adjust the Integration Limits
Since we changed variables to \( u = 3v + 1 \), we need new bounds for \( u \). When \( v = 1 \), \( u = 3(1) + 1 = 4 \). When \( v = 5 \), \( u = 3(5) + 1 = 16 \). Thus, the integral becomes \( 3 \int_{4}^{16} \sqrt{u} \, du \).
04
Integrate with Respect to u
The integral \( \int \sqrt{u} \, du \) can be written as \( \int u^{1/2} \, du \). The antiderivative of \( u^{1/2} \) is \( \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2} \). Substituting back, our integral becomes \( 3 \cdot \left[ \frac{2}{3} u^{3/2} \right]_{4}^{16} \).
05
Evaluate the Definite Integral
Now calculate \( 3 \cdot \left[ \frac{2}{3} (16)^{3/2} - \frac{2}{3} (4)^{3/2} \right] \). Simplify this to \( 3 \cdot \left[ \frac{2}{3} (64) - \frac{2}{3} (8) \right] \), which becomes \( 3 \cdot \frac{2}{3} \cdot (64 - 8) = 3 \cdot \frac{2}{3} \cdot 56 = 2 \cdot 56 = 112 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful technique in calculus for simplifying complex integrals. When faced with a function composed of a composite or nested term, substitution can turn it into an easier expression.
In this approach, we replace the complex inside function with a new variable, often denoted as \( u \). This transforms the original integral into one in terms of \( u \), which is hopefully simpler to evaluate.
In this approach, we replace the complex inside function with a new variable, often denoted as \( u \). This transforms the original integral into one in terms of \( u \), which is hopefully simpler to evaluate.
- Choose a substitution: In this exercise, we let \( u = 3v + 1 \), which simplifies \( 9 \sqrt{3v + 1} \) into \( 9 \sqrt{u} \).
- Find the differential \( du \): Since \( du/dv = 3 \), solving gives us \( dv = \frac{1}{3} du \).
- Change the entire integral: Replace \( dv \) with \( \frac{1}{3} du \), turning the integral into \( 3 \int \sqrt{u} \, du \).
Integration Limits
When performing integration via substitution, it is crucial not to forget the integration limits. Any change of variable requires a corresponding update to the bounds of the integral.
For the substitution \( u = 3v + 1 \), we adjusted the limits from \( v \)-terms to \( u \)-terms:
For the substitution \( u = 3v + 1 \), we adjusted the limits from \( v \)-terms to \( u \)-terms:
- Lower limit: When \( v = 1 \), substituting gives \( u = 3(1) + 1 = 4 \).
- Upper limit: When \( v = 5 \), substituting gives \( u = 3(5) + 1 = 16 \).
Antiderivative
Finding the antiderivative, or the indefinite integral, is a key step in evaluating definite integrals. The antiderivative involves reversing the process of differentiation.
In this exercise, we need the antiderivative of \( \sqrt{u} \), which is expressed as \( u^{1/2} \). Using the power rule for integration:
In this exercise, we need the antiderivative of \( \sqrt{u} \), which is expressed as \( u^{1/2} \). Using the power rule for integration:
- The power rule states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where \( n eq -1 \).
Evaluate Integrals
After finding the antiderivative, the final task in evaluating definite integrals is to compute the difference at the upper and lower bounds.
Our transformed integral becomes:\[ 3 \cdot \left[ \frac{2}{3} u^{3/2} \right]_4^{16} \]Let's break it down:
Our transformed integral becomes:\[ 3 \cdot \left[ \frac{2}{3} u^{3/2} \right]_4^{16} \]Let's break it down:
- Evaluate at upper limit \( u = 16 \): \( \frac{2}{3} (16)^{3/2} = \frac{2}{3} \cdot 64 \).
- Evaluate at lower limit \( u = 4 \): \( \frac{2}{3} (4)^{3/2} = \frac{2}{3} \cdot 8 \).
- Find the difference: \[ \frac{2}{3} \cdot (64 - 8) = \frac{2}{3} \cdot 56 \].
- Multiply by 3: \( 3 \cdot \frac{2}{3} \cdot 56 = 2 \cdot 56 = 112 \).
