Difference between revisions of "009B Sample Final 3, Problem 7"

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!Step 1:    
 
!Step 1:    
 
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|We use the Direct Comparison Test for Improper Integrals.
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|For all &nbsp;<math>x</math>&nbsp; in &nbsp;<math>[1,\infty),</math>
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>0\le \frac{\sin^2(x)}{x^3} \le \frac{1}{x^3}.</math>
 
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|Also,
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{\sin^2(x)}{x^3}</math>&nbsp; and &nbsp;<math>\frac{1}{x^3}</math>
 
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|are continuous on &nbsp;<math>[1,\infty).</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Now, we have
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{\int_1^\infty \frac{1}{x^3}~dx} & = & \displaystyle{\lim_{a\rightarrow \infty} \int_1^a \frac{1}{x^3}~dx}\\
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&&\\
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& = & \displaystyle{\lim_{a\rightarrow \infty} \frac{1}{-2x^2}\bigg|_1^a}\\
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&&\\
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& = & \displaystyle{\lim_{a\rightarrow \infty} \frac{1}{-2a^2}+\frac{1}{2}}\\
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&&\\
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& = & \displaystyle{\frac{1}{2}.}
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\end{array}</math>
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|-
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|Since &nbsp;<math>\int_1^\infty \frac{1}{x^3}~dx</math>&nbsp; converges,
 
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|  
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math>
 
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|converges by the Direct Comparison Test for Improper Integrals.
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|&nbsp; &nbsp; &nbsp; &nbsp;converges
 
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[[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 10:12, 2 March 2017

Does the following integral converge or diverge? Prove your answer!

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_1^\infty \frac{\sin^2(x)}{x^3}~dx}
Foundations:  
Direct Comparison Test for Improper Integrals
        Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}   be continuous on  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,\infty)}
        where  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\le f(x)\le g(x)}   for all  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}   in  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,\infty).}
       1.  If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^\infty g(x)~dx}   converges, then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^\infty f(x)~dx}   converges.
       2.  If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^\infty f(x)~dx}   diverges, then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^\infty g(x)~dx}   diverges.


Solution:

Step 1:  
We use the Direct Comparison Test for Improper Integrals.
For all  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}   in  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [1,\infty),}
        Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\le \frac{\sin^2(x)}{x^3} \le \frac{1}{x^3}.}
Also,
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sin^2(x)}{x^3}}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{x^3}}
are continuous on  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [1,\infty).}
Step 2:  
Now, we have
        Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\int_1^\infty \frac{1}{x^3}~dx} & = & \displaystyle{\lim_{a\rightarrow \infty} \int_1^a \frac{1}{x^3}~dx}\\ &&\\ & = & \displaystyle{\lim_{a\rightarrow \infty} \frac{1}{-2x^2}\bigg|_1^a}\\ &&\\ & = & \displaystyle{\lim_{a\rightarrow \infty} \frac{1}{-2a^2}+\frac{1}{2}}\\ &&\\ & = & \displaystyle{\frac{1}{2}.} \end{array}}
Since  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_1^\infty \frac{1}{x^3}~dx}   converges,
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_1^\infty \frac{\sin^2(x)}{x^3}~dx}
converges by the Direct Comparison Test for Improper Integrals.


Final Answer:  
       converges

Return to Sample Exam

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