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Thread: Calculate the volume of the solid

  1. #1
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    Calculate the volume of the solid

    Hey guys, so this topic was just introduced this week and attached, I have attached the question (#48) and what my teacher wants me to answer on it, just to be specific, I just want to know how to do a) and then I will struggle with the rest later (hopefully not struggle but we will see). I have also attached what I have so far. Any help would be greatly appreciated! Thanks!
    Attached Thumbnails Attached Thumbnails Calculate the volume of the solid-img_0506.jpg   Calculate the volume of the solid-img_0507.jpg   Calculate the volume of the solid-img_0508.jpg  
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  2. #2
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    Re: Calculate the volume of the solid

    Turn the frustrum sideways such that the center of the circular top is at the origin (see ruff sketch).
    The slanted side of the frustrum would be the line $y=\dfrac{R-r}{h}x + r$ ... this is the line that will be rotated about the x-axis from x = 0 to x = h.

    $\displaystyle V = \pi \int_0^h \left(\dfrac{R-r}{h}x + r\right)^2 \, dx$
    Attached Thumbnails Attached Thumbnails Calculate the volume of the solid-img_1590.png  
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    Re: Calculate the volume of the solid

    The formula makes sense but I am a little unclear as to how you solved for the radius of the circle. Sorry, thanks again!
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  4. #4
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    Re: Calculate the volume of the solid

    if you're referring to the disk's radius of rotation, it's just the distance from the x-axis to the linear function ... $r(x) = \dfrac{R-r}{h}x + r$

    $\displaystyle V = \pi \int_a^b [r(x)]^2 \, dx$
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