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Thread: Exponential Integral Questions

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    Exponential Integral Questions

    1. The curve y = √(ex +1) is rotated about the x-axis from x = 0 to x = 1. Find the exact volume of the solid formed.

    2. Find the exact area enclosed between the curve y = e2x and the lines y = 1 and x = 2.

    Thanks

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    Re: Exponential Integral Questions

    1)
    y^2 = e^x + 1
    Volume = pi* integral(0 to 1) (e^x + 1) dx
    = pi* [e^x + x] (0 to 1)
    = pi* (e + 1) - e^0
    = pi*e units cubed

    2) At y = 1 we have 1 = e^2x
    ln1 = 2x
    x = 0

    So we integrate from 0 to 2
    A = integral(0 to 2) (e^2x) dx
    = [e^2x / 2] (0 to 2)
    = e^4 / 2 - 1/2
    = 1/2 (e^4 - 1) units squared

    Edit: yeah minus the rectangle as 1729 said
    Edit: I got my second answer off yahoo answers (not copy pasted but I copied their answer) rip got betrayed
    Last edited by pikachu975; 28 Jan 2017 at 3:41 AM.
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    Re: Exponential Integral Questions

    Quote Originally Posted by pikachu975 View Post
    1)
    y^2 = e^x + 1
    Volume = pi* integral(0 to 1) (e^x + 1) dx
    = pi* [e^x + x] (0 to 1)
    = pi* (e + 1) - e^0
    = pi*e units cubed

    2) At y = 1 we have 1 = e^2x
    ln1 = 2x
    x = 0

    So we integrate from 0 to 2
    A = integral(0 to 2) (e^2x) dx
    = [e^2x / 2] (0 to 2)
    = e^4 / 2 - 1/2
    = 1/2 (e^4 - 1) units squared
    Thanks for your reply!!

    Yeah I got the same answer for question 1 but I thought the textbook had a different answer - appears it was right all along! (my bad)

    I got the same for question 2 as well but the textbook answers say 1/2(e^4 - 5) units squared. (I'm sure it says that)

    Thoughts?
    Last edited by boredofstudiesuser1; 28 Jan 2017 at 12:43 AM.

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    Yeah I got the same answer for question 1 but I thought the textbook had a different answer - appears it was right all along! (my bad)

    I got the same for question 2 as well but the textbook answers say 1/2(e^4 - 5) units squared. (I'm sure it says that)

    Thoughts?
    Last edited by 1729; 28 Jan 2017 at 10:00 PM.
    pikachu975 likes this.

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    Re: Exponential Integral Questions

    Quote Originally Posted by 1729 View Post
    The textbook is correct.

    Thanks!

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    Re: Exponential Integral Questions

    Two more questions:

    1. Find stat points and inflexions for the curve y=(In(x) - 1)3 and determine their nature.

    2.(3x+3)/(x2-9) = 1/(x+3) + 2/(x-3)

    Hence find ∫(3x+3)/(x2-9) dx

    EDIT: For question number 2, I got:

    In(x) + (1/2)In(x-3) + C

    The answer says 2 instead of (1/2)
    Last edited by boredofstudiesuser1; 28 Jan 2017 at 7:58 PM.

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    Two more questions:

    1. Find stat points and inflexions for the curve y=(In(x) - 1)3 and determine their nature.

    2.(3x+3)/(x2-9) = 1/(x+3) + 2/(x-3)

    Hence find ∫(3x+3)/(x2-9) dx

    EDIT: For question number 2, I got:

    In(x) + (1/2)In(x-3) + C

    The answer says 2 instead of (1/2)
    It is 2.




    Last edited by 1729; 25 Feb 2017 at 12:27 AM.

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    Two more questions:

    1. Find stat points and inflexions for the curve y=(In(x) - 1)3 and determine their nature.

    2.(3x+3)/(x2-9) = 1/(x+3) + 2/(x-3)

    Hence find ∫(3x+3)/(x2-9) dx




    Can you get it from here? (If you're not sure about how to solve the equations f'(x) = 0 and f"(x) = 0, you can use an intermediary substitution u = ln(x) – 1.)

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    Re: Exponential Integral Questions

    Quote Originally Posted by 1729 View Post
    It is 2
    Yes, I would assume the answer is right but I'm unsure as to what I did wrong...

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    Re: Exponential Integral Questions

    Quote Originally Posted by InteGrand View Post




    Can you get it from here?
    Based on that, I got (e,0) as a min TP but apparently it's a point of inflexion?
    Last edited by boredofstudiesuser1; 28 Jan 2017 at 8:06 PM.

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    Yes, I would assume the answer is right but I'm unsure as to what I did wrong...
    It's a 2, because a primitive of 2/(x – 3) is 2*ln|x – 3|.

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    Re: Exponential Integral Questions

    Quote Originally Posted by InteGrand View Post
    It's a 2, because a primitive of 2/(x – 3) is 2*ln|x – 3|.
    Oh right, you need a 2 to make 1 equal 2 *facepalm*

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    Based on that, I got (e,0) as a min TP but apparently it's a point of inflexion?
    How did you find it to be a local minimum?

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    Re: Exponential Integral Questions

    Quote Originally Posted by InteGrand View Post
    How did you find it to be a local minimum?
    With the first derivative test

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    Based on that, I got (e,0) as a min TP but apparently it's a point of inflexion?


    Quote Originally Posted by boredofstudiesuser1 View Post
    With the first derivative test
    Last edited by 1729; 28 Jan 2017 at 8:29 PM.

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    Re: Exponential Integral Questions

    Quote Originally Posted by 1729 View Post
    Yeah it appears to work that way.

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    With the first derivative test
    If you use the first derivative test, you should be able to see that f'(x) is positive just to the left of e and also just to the right of e, so it's neither a local minimum nor a maximum. It is a stationary point of inflection.

    (Note that the numerator of f'(x) is always positive for x in the domain and x not equal to e, so the sign of f'(x) is just the sign of the denominator (which is x), so in any neighbourhood of e, we have f'(x) > 0, for x not equal to e.)

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    Re: Exponential Integral Questions

    This is a concept based question.

    If I need to integrate a normal exponential with the base being a number other than e, can I follow this formula?:

    ∫n^(ax) = (n^ax)/a(loge(n))

    If not, could anyone provide a sorta formula you could use for integrating logarithms and exponentials with a base other than e?
    Last edited by boredofstudiesuser1; 28 Jan 2017 at 8:34 PM.

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    This is a concept based question.

    If I need to integrate a normal exponential with the base being a number other than e, can I follow this formula?:

    ∫n^(ax) = (n^ax)/a(loge(n))

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    This is a concept based question.

    If I need to integrate a normal exponential with the base being a number other than e, can I follow this formula?:

    ∫n^(ax) = (n^ax)/a(loge(n))
    Last edited by 1729; 25 Feb 2017 at 12:26 AM.

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    Re: Exponential Integral Questions

    Quote Originally Posted by 1729 View Post
    How would I do this for a logarithm with a base other than e?

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    How would I do this for a logarithm with a base other than e?
    What do you mean? (Do what?)

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    Re: Exponential Integral Questions

    Quote Originally Posted by InteGrand View Post
    What do you mean? (Do what?)
    Is there a formula I can follow for integrating logarithms with a base other than e?
    Last edited by boredofstudiesuser1; 28 Jan 2017 at 8:51 PM.

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    Re: Exponential Integral Questions

    Quote Originally Posted by boredofstudiesuser1 View Post
    Is there a formula I can follow for integrating logarithms with a base other than e?
    What you were asking for before was integrating exponentials rather than logarithms. The way to do this has been shown above by 1729 and you also gave a formula for it.

    Integrating logarithms from scratch is not within the realms of 2U (only 4U).

    Also you only would need to know how to do these for base e logarithms, because you can convert an arbitrary-based logarithm into log base e by a simple rescaling (use change-of-base formula).

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    Re: Exponential Integral Questions

    Quote Originally Posted by InteGrand View Post
    What you were asking for before was integrating exponentials rather than logarithms. The way to do this has been shown above by 1729 and you also gave a formula for it.

    Integrating logarithms from scratch is not within the realms of 2U (only 4U).

    Also you only would need to know how to do these for base e logarithms, because you can convert an arbitrary-based logarithm into log base e by a simple rescaling (use change-of-base formula).
    Right ok, thank you very much for all your help.

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