Improper Integral

  • Thread starter darkchild
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  • #1
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Homework Statement


[tex]\int_{-3}^{1}\frac{x}{\sqrt{9-x^{2}}}[/tex]

Homework Equations



Let f be continuous on the half-open interval (a, b] and suppose that
[tex] \lim_{x \to a^{+}} |f(x)| = \infty[/tex]. Then

[tex]\int_{a}^{b}f(x) dx = \lim_{ t \to a^{+}}\int_{t}^{b}f(x)
dx[/tex]



The Attempt at a Solution



[tex]\int_{-3}^{1}\frac{x}{\sqrt{9-x^{2}}}

= \lim_{ t \to -3^{+}}\int_{t}^{1}\frac{x}{\sqrt{9-x^{2}}}

[/tex]

[tex] u = 9 - x^{2} [/tex]
[tex] du = -2x dx [/tex]

[tex] \lim_{ t \to -3^{+}}\int_{t}^{1}\frac{x}{\sqrt{9-x^{2}}}

= \lim_{ t \to 0^{+}}-\frac{1}{2}\int_{t}^{8} u^{-1/2} du

=\lim_{ t \to 0^{+}}-u^{1/2}|_{t}^{8}

=-\sqrt{3sin(1)} + \lim_{ t \to -3^{+}}\sqrt{3sin(t)}

=-1.588840129 + ?[/tex]

I get 0 for the limit, but according to Maple and my graphing calculator, that does not give the correct value for this integral. The correct value is approximately -2.8. May I please have some guidance as to what may have went wrong?
 

Answers and Replies

  • #2
40
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[tex]\int_{-3}^{1}\frac{x}{\sqrt{9-x^{2}}}

= \lim_{ t \to -3^{+}}\int_{t}^{1}\frac{x}{\sqrt{9-x^{2}}}

[/tex]

[tex] u = 9 - x^{2} [/tex]
[tex] du = -2x dx [/tex]

[tex] \lim_{ t \to -3^{+}}\int_{t}^{1}\frac{x}{\sqrt{9-x^{2}}}

= \lim_{ t \to 0^{+}}-\frac{1}{2}\int_{t}^{8} u^{-1/2} du

=\lim_{ t \to 0^{+}}-u^{1/2}|_{t}^{8}

=-\sqrt{3sin(1)} + \lim_{ t \to -3^{+}}\sqrt{3sin(t)}

=-1.588840129 + ?[/tex]

I get 0 for the limit, but according to Maple and my graphing calculator, that does not give the correct value for this integral. The correct value is approximately -2.8. May I please have some guidance as to what may have went wrong?

I don't understand that step where you introduce sine. You should just get:

[tex]
\lim_{ t \to 0^{+}}-u^{1/2}|_{t}^{8}

=-\lim_{t \to 0^+}(\sqrt{8} - \sqrt{t})

= 0 - 2\sqrt{2}

= -2\sqrt{2}

\approx -2.8
[/tex]
 
  • #3
155
0
I don't understand that step where you introduce sine. You should just get:

Oh, god, I made an incredibly stupid mistake....Thank you.
 
  • #4
40
0
haha, it happens. no worries.
 

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