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    Compute the triple integral F(x,y,z) = z over the region R in the first octant bounded by the planes y = 0, z = 0, x + y = 2, 2y + x = 6, and the cylinder y^2 + z^2 = 4. (Answer 26/3).

    This is what I have so far

    \int_{a}^{b} \int_{a}^{b} \int_{0}^{\sqrt{4-y^{2}}} Z dzdxdy

    Any tips how to get the next limits I mean I can't sketch this so dunno these questions are frustrating for me. Especially since I haven't done 3d geometry since high school.
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    (Original post by CHEN20041)
    Compute the triple integral F(x,y,z) = z over the region R in the first octant bounded by the planes y = 0, z = 0, x + y = 2, 2y + x = 6, and the cylinder y^2 + z^2 = 4. (Answer 26/3).

    This is what I have so far

    \int_{a}^{b} \int_{a}^{b} \int_{0}^{\sqrt{4-y^{2}}} Z dzdxdy

    Any tips how to get the next limits I mean I can't sketch this so dunno these questions are frustrating for me. Especially since I haven't done 3d geometry since high school.
    Note that the cylinder has the x-axis as its axis of rotational symmetry, and so with the radius being 2, you are interested in the region 0<y<2 and 0<z<2.

    Your z limit is fine for this. Then for the x limit, just consider the xy plane sketch of x+y=2 and 2y+x=6 and deduce your x limit there. Finish it off with the y limit. Can be tempting to say that the y limit is 0 to 3 from this sketch, but remember but we have it 0<y<2 in order to stay within the cylinder.
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    (Original post by RDKGames)
    Note that the cylinder has the x-axis as its axis of rotational symmetry, and so with the radius being 2, you are interested in the region 0&lt;y&lt;2 and 0&lt;z&lt;2.

    Your z limit is fine for this. Then for the x limit, just consider the xy plane sketch of x+y=2 and 2y+x=6 and deduce your x limit there. Finish it off with the y limit. Can be tempting to say that the y limit is 0 to 3 from this sketch, but remember but we have it 0<y<2 in order to stay within the cylinder.
    Cheers mate nice
 
 
 
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