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Math Help - Integration (finding the are bounded by a curve)

  1. #1
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    Integration (finding the are bounded by a curve)

    To find the are bounded by the curve

    F(x) = xe^(x^2) and the limits a=-2 and b=-1

    So I integrate the function using substitution

    u=x and get stuck on how to go on

    Help much appreciated!! thanks!!!
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  2. #2
    Member mathemagister's Avatar
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    Quote Originally Posted by Yehia View Post
    To find the are bounded by the curve

    F(x) = xe^(x^2) and the limits a=-2 and b=-1

    So I integrate the function using substitution

    u=x and get stuck on how to go on

    Help much appreciated!! thanks!!!
    I think you are on the wrong track. Look at the function:

    xe^{x^2}

    Now integrate it simply, don't use any U-Substitution.

    \int{xe^{x^2}}dx = \frac{1}{2}e^{x^2} + C

    I'll let you take it from there. If you still don't know how to do it, feel free to ask me.
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  3. #3
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    Quote Originally Posted by mathemagister View Post
    I think you are on the wrong track. Look at the function:

    xe^{x^2}

    Now integrate it simply, don't use any U-Substitution.

    \int{xe^{x^2}}dx = \frac{1}{2}e^{x^2} + C

    I'll let you take it from there. If you still don't know how to do it, feel free to ask me.

    Aaahh no i see it now. Thanks a lot for your help. I see now why. NO substitution required.
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  4. #4
    Member mathemagister's Avatar
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    Quote Originally Posted by Yehia View Post
    Aaahh no i see it now. Thanks a lot for your help. I see now why. NO substitution required.
    Anytime
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  5. #5
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    Quote Originally Posted by mathemagister View Post
    Anytime
    Hi, sorry could you help me with just one more problem?
    If i wish to find the Volume of a curve rotated round the x axis. i know it is the integral of pix^2, but in the funtion: f(x)=x^(1/2)e^x with the limits 1 and 2, i am stuck
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  6. #6
    Member mathemagister's Avatar
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    Quote Originally Posted by Yehia View Post
    Hi, sorry could you help me with just one more problem?
    If i wish to find the Volume of a curve rotated round the x axis. i know it is the integral of pix^2, but in the funtion: f(x)=x^(1/2)e^x with the limits 1 and 2, i am stuck
    The formula for the volume of any function rotated around y=k is given by V=\int_a^b \pi (R^2 - r^2) dx where a and b are the limits of integration, R (outer radius) is the distance from the axis of rotation to the further side of the elemental strip and r (inner radius) is the distance from the axis of rotation to the closer side of the elemental strip. In your case, since the axis of rotation is the x-axis, which is y=0, your r is simply 0. This will cancel out and your formula will become V=\int_a^b \pi R^2 dx.

    Your R (outer radius) is simply the function because your axis of rotation is the x-axis.

    Your answer becomes:
    V=\int_1^2 \pi (f(x))^2 dx

    Hope I helped
    Do you get it now?
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  7. #7
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    Quote Originally Posted by mathemagister View Post
    The formula for the volume of any function rotated around y=k is given by V=\int_a^b \pi (R^2 - r^2) dx where a and b are the limits of integration, R (outer radius) is the distance from the axis of rotation to the further side of the elemental strip and r (inner radius) is the distance from the axis of rotation to the closer side of the elemental strip. In your case, since the axis of rotation is the x-axis, which is y=0, your r is simply 0. This will cancel out and your formula will become V=\int_a^b \pi R^2 dx.

    Your R (outer radius) is simply the function because your axis of rotation is the x-axis.

    Your answer becomes:
    V=\int_1^2 \pi (f(x))^2 dx

    Hope I helped
    Do you get it now?
    Yeah I get how to do it normally, it's just this specific one that i don't get.

    the function is (x^0.5)(e^x) and the limits are 1 and 2 :S
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  8. #8
    Member mathemagister's Avatar
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    Quote Originally Posted by Yehia View Post
    Yeah I get how to do it normally, it's just this specific one that i don't get.

    the function is (x^0.5)(e^x) and the limits are 1 and 2 :S
    Oh you wrote the function weirdly before with LaTex and I wasn't sure what you meant.

    \int_1^2 \pi e^x \sqrt{x} dx

    Well, this is actually a non-elementary integral. This is impossible to compute normally. If you are more advanced in calculus you can do it, though. Do you know Dawson's function?

    Well the answer to the indefinite integral is e^x(\sqrt{x}-F(\sqrt{x}) + C where F is the Dawson Function. Yeah, a first (or even second) year calc student would not be responsible for being able to do this integral. Are you sure you weren't allowed to use a graphing calculator?
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