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Math Help - Calculating the following integrals

  1. #1
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    Calculating the following integrals

    Hullo,

    I need some help calculating the following integrals:

     \lim_{x\to 0}\; \dfrac{1}{x} \int_0^x\; e^{t^2}\; dt<br />

    and this one:

     \lim_{h\to 0}\; \dfrac{1}{h} \int_3^{3+h}\; e^{t^2}\; dt<br />

    using the FTC.

    Any help would be appreciated.

    -the Doctor
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  2. #2
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    Quote Originally Posted by thedoctor818 View Post
    Hullo,

    I need some help calculating the following integrals:

     \lim_{x\to 0}\; \dfrac{1}{x} \int_0^x\; e^{t^2}\; dt<br />

    and this one:

     \lim_{h\to 0}\; \dfrac{1}{h} \int_3^{3+h}\; e^{t^2}\; dt<br />

    using the FTC.


    Any help would be appreciated.

    -the Doctor
    Both these problems have very similar solutions.

    Here's a way to start

    Let  F(x) = \int_0^{x}\; e^{t^2}\; dt

    Notice that (for part 2)  \frac{F(x+h) - F(x)}{h} = \frac{\int_0^{x+h}\; e^{t^2}\; dt  - \int_0^{x}\; e^{t^2}\; dt}{h} = \frac{1}{h}\int_x^{x+h}\; e^{t^2}\; dt ...
    Last edited by southprkfan1; April 20th 2010 at 10:27 AM.
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  3. #3
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    I am sorry, but am still a bit confused. So, for the first integral, am I just taking  lim_{x\to0}\dfrac{F(x)}{x} ? If so , how do I evaluate since there is no 'elementary' antiderivative. And in the second, am I taking  lim_{h\to0}\dfrac{F(x+h)-F(x)}{h} ?

    Sorry, but I am still somewhat confused.

    -Michael
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  4. #4
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    Quote Originally Posted by thedoctor818 View Post
    And in the second, am I taking  lim_{h\to0}\dfrac{F(x+h)-F(x)}{h} ?
    Yes, that should look like something familiar.

    So, for the first integral, am I just taking
    Yes, but note that F(0) = 0, so we can rewrite it as:

     \dfrac{F(x)}{x} = \dfrac{F(x)-F(0)}{x-0}
    Last edited by southprkfan1; April 20th 2010 at 10:27 AM.
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  5. #5
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    Thanks a bunch - that really helped out.

    -Michael
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