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

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|'''Direct Comparison Test for Improper Integrals'''
 
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|&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math style="vertical-align: -5px">f</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">g</math>&nbsp; be continuous on &nbsp;<math style="vertical-align: -5px">[a,\infty)</math>
 
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|&nbsp; &nbsp; &nbsp; &nbsp; where &nbsp;<math style="vertical-align: -5px">0\le f(x)\le g(x)</math>&nbsp; for all &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; in &nbsp;<math style="vertical-align: -5px">[a,\infty).</math>
 
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|&nbsp; &nbsp; &nbsp; &nbsp;'''1.'''&nbsp; If &nbsp;<math style="vertical-align: -14px">\int_a^\infty g(x)~dx</math>&nbsp; converges, then &nbsp;<math style="vertical-align: -14px">\int_a^\infty f(x)~dx</math>&nbsp; converges.
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|&nbsp; &nbsp; &nbsp; &nbsp;'''2.'''&nbsp; If &nbsp;<math style="vertical-align: -14px">\int_a^\infty f(x)~dx</math>&nbsp; diverges, then &nbsp;<math style="vertical-align: -14px">\int_a^\infty g(x)~dx</math>&nbsp; diverges.
 
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Revision as of 09:21, 1 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:  
Step 2:  


Final Answer:  

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