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alffer1

  • 11 months ago

Multivariable integration help: Compute the moment of inertia around the y-axis associated with the region |y|<x^2, |x|<2, assuming constant density delta = 5/32. Not sure what to start with...

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  1. oldrin.bataku
    • 11 months ago
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    the moment of inertia \((I_x,I_y)\) is computed using:$$I_x=\iint_R x^2 \delta(x,y)\ dx\ dy\\I_y=\iint_R y^2\delta(x,y)\ dx\ dy$$

  2. oldrin.bataku
    • 11 months ago
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    here \(R\) is given by \(-x^2<y<x^2\) and \(-2<x<2\) whereas \(\delta=5/32\) ergo:$$\begin{align*}I_x&=\int_{-2}^2\int_{-x^2}^{x^2}x^2\cdot\frac5{32}\ dy\ dx\\&=\frac5{32}\int_{-2}^2 x^2\int_{-x^2}^{x^2}\ dy\ dx\\&=\frac5{32}\int_{-2}^2 2x^4\ dx\\&=\frac18\int_0^25x^4\ dx\\&=\frac18(2^5-0^5)\\&=4\end{align*}$$

  3. oldrin.bataku
    • 11 months ago
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    can you find \(I_y\)?

  4. alffer1
    • 11 months ago
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    oh duh...I forgot I had 2 dimensions to work with. Thanks a ton!

  5. oldrin.bataku
    • 11 months ago
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    no problem :-p it's the same integrand only \(y^2\)

  6. alffer1
    • 11 months ago
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    wait so same as the first line in the bounds too?

  7. oldrin.bataku
    • 11 months ago
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    indeed

  8. alffer1
    • 11 months ago
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    just a sec, somewhat confused, are the bounds changed to y^2? no, right?

  9. oldrin.bataku
    • 11 months ago
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    nope!

  10. alffer1
    • 11 months ago
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    ok got the answers. you were great. Saved my neck here.

  11. oldrin.bataku
    • 11 months ago
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    $$I_y=\int_{-2}^2\int_{-x^2}^{x^2}y^2\cdot\frac5{32}\ dy\ dx=\frac5{32}\int_{-2}^2\left[\frac13y^3\right]_{-x^2}^{x^2}\ dx=\frac5{32}\int_{-2}^2\frac23x^6dx$$so we get:$$I_y=\frac5{48}\int_{-2}^2 x^6\ dx=\frac5{24}\int_0^2x^6\ dx=\frac5{24}\cdot\frac17(2^7-0^7)=\frac{5\cdot16}{3\cdot7}=\frac{80}{21}$$

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