Difference between revisions of "009B Sample Final 3, Problem 7"
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|We use the Direct Comparison Test for Improper Integrals. | |We use the Direct Comparison Test for Improper Integrals. | ||
|- | |- | ||
| − | |For all <math>x</math> in <math>[1,\infty),</math> | + | |For all <math style="vertical-align: 0px">x</math> in <math style="vertical-align: -5px">[1,\infty),</math> |
|- | |- | ||
| <math>0\le \frac{\sin^2(x)}{x^3} \le \frac{1}{x^3}.</math> | | <math>0\le \frac{\sin^2(x)}{x^3} \le \frac{1}{x^3}.</math> | ||
| Line 31: | Line 31: | ||
|Also, | |Also, | ||
|- | |- | ||
| − | | <math>\frac{\sin^2(x)}{x^3}</math> and <math>\frac{1}{x^3}</math> | + | | <math style="vertical-align: -15px">\frac{\sin^2(x)}{x^3}</math> and <math style="vertical-align: -15px">\frac{1}{x^3}</math> |
|- | |- | ||
| − | |are continuous on <math>[1,\infty).</math> | + | |are continuous on <math style="vertical-align: -5px">[1,\infty).</math> |
|} | |} | ||
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\end{array}</math> | \end{array}</math> | ||
|- | |- | ||
| − | |Since <math>\int_1^\infty \frac{1}{x^3}~dx</math> converges, | + | |Since <math style="vertical-align: -15px">\int_1^\infty \frac{1}{x^3}~dx</math> converges, |
|- | |- | ||
| <math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math> | | <math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math> | ||
Revision as of 10:17, 2 March 2017
Does the following integral converge or diverge? Prove your answer!
| Foundations: |
|---|
| Direct Comparison Test for Improper Integrals |
| Let and 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 |