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Thread: Cylinder

  1. #1
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    Cylinder

    Let r be a positive constant. Consider the cylinder $\displaystyle x^2+y^2≤r^2$ and let C be the part of the cylinder that satisfies 0 ≤ z ≤ y.

    (3) Let a be the length of the arc along the base circle of C from the point (r, 0, 0) to the point (r cos θ, r sin θ, 0) (0 ≤ θ ≤ π). Let b be the length of the line segment from the point (r cos θ, r sin θ, 0) to the point (r cos θ, r sin θ, r sin θ). Express a and b in terms of r, θ.

    ( 4) Calculate the area of the side of C with x2+y2 = r2, and express it in terms of r. 2. Relevant equations Not sure 3. The attempt at a solution

    I used the formula . [f '(x)] = x/(r-x) ... r∫π0dθ=??? the answer ir θr
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  2. #2
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    Re: Cylinder

    Cylinder-screenshot_20180330-135550.png
    Attached Thumbnails Attached Thumbnails Cylinder-image045.gif  
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  3. #3
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    Re: Cylinder

    $$x^2+y^2=r^2$$

    $$2x + 2y \dfrac{dy}{dx} = 0$$

    $$\dfrac{dy}{dx} = -\dfrac{x}{y}$$

    $$\dfrac{dy}{dx} = \dfrac{x^2}{y^2}$$

    So, your formula is $L = \int_r^{r\cos \theta} \sqrt{1+\dfrac{x^2}{y^2}}dx$

    Next, we have $x = r\cos \theta, dx = -r\sin \theta d\theta, y = r\sin \theta$, and at $x=r$, $\theta = 0$ and at $x=r\cos \theta$, $\theta = \theta$

    $$L = \int_r^{r\cos \theta} \sqrt{1+\dfrac{x^2}{y^2}} dx = r\int_0^\theta \sqrt{1+\cot^2 \theta} \sin \theta d\theta = r\int_0^\theta d\theta = r\theta$$
    Thanks from helpprovaexata
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  4. #4
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    Re: Cylinder

    thanks a lot!!!!

    why you substituted dx=−rsinθdθ for dx=rsinθdθ? The -r changed for +r?
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  5. #5
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    Re: Cylinder

    Quote Originally Posted by helpprovaexata View Post
    thanks a lot!!!!

    why you substituted dx=−rsinθdθ for dx=rsinθdθ? The -r changed for +r?
    I typed it from my phone. Arc length is always positive. I missed the absolute value signs.
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  6. #6
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    Re: Cylinder

    in the second part the z-axis varied rsinθ so just put this?

    the answer is rsinθ
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  7. #7
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    Re: Cylinder

    Quote Originally Posted by helpprovaexata View Post
    in the second part the z-axis varied rsinθ so just put this?

    the answer is rsinθ
    Correct
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  8. #8
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    Re: Cylinder

    How would i handle the integral of question d?
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  9. #9
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    Re: Cylinder

    ??What "question d"? Do you mean question 4, "Calculate the area of the side of C with $x^2+y^2 = r^2$"?
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  10. #10
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    Re: Cylinder

    Yes, number 4. Sorry
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  11. #11
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    Re: Cylinder

    Basically, in part 3, we found a map from $(x,y,z)$ to $(r, \theta)$. The base of the cylinder is $r\theta$ while the height is $r\sin \theta$. You have $0< \theta < \pi$ (because for the rest, $z=0$). So, you are taking the integral $\displaystyle \int_0^\pi r^2\sin \theta d\theta = 2r^2$.
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