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Math Help - Rotating around x and y axis

  1. #1
    Member Jones's Avatar
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    Rotating around x and y axis

    Hi,

    I have a problem i wasn't able to solve, i don't know how to calculate the volume of the generated body, when it is rotated around both the x and the y axis.

     A = \{(x,y):\sqrt{x}e^{x^2}~\le y \le 3,~ 0 \le x \le 1 \}

    Jones
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  2. #2
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    Quote Originally Posted by Jones View Post
    Hi,

    I have a problem i wasn't able to solve, i don't know how to calculate the volume of the generated body, when it is rotated around both the x and the y axis.

     A = \{(x,y):\sqrt{x}e^{x^2}~\le y \le 3,~ 0 \le x \le 1 \}

    Jones
    I can show you how to calculate the volume when the curve rotates about the x-axis.

    The general formula for a rotation volume with the x-axis as axis of rotation is:

    V_{rotx} = \pi\int_a^b y^2 dx

    Plug in the terms you know:

    V_{rotx}=\pi \int_0^1\left(\sqrt{x} \cdot e^{x^2} \right)^2 dx

    V_{rotx}=\pi \int_0^1\left(x \cdot e^{2x^2} \right) dx

    Now use integration by substitution:

    u = 2x^2~\implies~\frac{du}{dx} = 4x~\implies~du = 4x dx

    The integral becomes:

    V_{rotx}=\frac{\pi}{4} \int_0^1\left(4x \cdot e^{2x^2} \right) dx

    V_{rotx}=\frac{\pi}{4} \int_0^1\left( e^{u} \right) du

    Can you take it from here?

    -----------------------------------------------------------------------

    There is missing at least one value to calculate the rotation volume if the y-axis is the axis of rotation.
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  3. #3
    Member Jones's Avatar
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    Quote Originally Posted by earboth View Post
    I can show you how to calculate the volume when the curve rotates about the x-axis.

    Can you take it from here?

    -----------------------------------------------------------------------

    There is missing at least one value to calculate the rotation volume if the y-axis is the axis of rotation.
    ?? You already did it for me. The only thing remaining is to plug in the limits.

    And what value is missing?, this question should be solvable
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  4. #4
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    Quote Originally Posted by Jones View Post
    ?? You already did it for me. The only thing remaining is to plug in the limits.

    And what value is missing?, this question should be solvable
    I've had some difficulties to understand this line:

    A = \{(x,y):\sqrt{x}e^{x^2}~\le y \le 3,~ 0 \le x \le 1 \}

    If you mean

    y = f(x)= \sqrt{x} \cdot e^{x^2}\ ,\ 0\leq x \leq 1

    then the property y \leq 3 is unnecessary because f(1) = e < 3.

    -----------------------------------------------

    To calculate the rotation volume with the y-axis as axis of rotation you have to use the general equation:

    V_{roty} = \int_a^b \pi \cdot x^2 dy

    Since \frac{dy}{dx} = y'~\implies~dy = y' \cdot dx

    this equation becomes:

    V_{roty} = \int_a^b \pi \cdot x^2 y' \cdot dx

    But unfortunately the integrand becomes very ugly so I can only provide you with an approximative result: V_{roty} \approx 3.994
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