Finding the volume under the plane x + y + z = 1My two methods are giving me different answers, and I'm not sure why.I am more confident in: ∫ 0 1 ∫ 0 1 − x ∫ 0 1 − x − y d z d y d x = 1 6 .But another way I'm trying is to sum the area of all the equilateral triangles underneath, and parallel to, the plane. The side lengths of these triangles range from 0 to 2 .Thus: ∫ 0 2 s 2 3 4 d s = 1 6 I am pretty sure the second approach is incorrect, but why?We are given x , y , z &gt; 0

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Finding the volume under the plane   x  +  y  +  z  =  1My two methods are giving me different answers, and I&#039;m not sure why.I am more confident in:       ∫    0    1        ∫    0          1      −      x            ∫    0          1      −      x      −      y        d  z     d  y     d  x  =      1    6  .But another way I&#039;m trying is to sum the area of all the equilateral triangles underneath, and parallel to, the plane. The side lengths of these triangles range from 0 to       2  .Thus:       ∫    0                  2                  s    2              3        4       d  s  =      1          6      I am pretty sure the second approach is incorrect, but why?We are given   x  ,  y  ,  z  &amp;gt;  0

ab8s1k28q · Accepted Answer

Step 1You are trying to do what is sometimes called &quot;volume-by-slicing&quot;. This is where you integrate the cross-sectional areas of the region to calculate the volume. When you do this, though, the cross-sections should be perpendicular to the &quot;axis of integration&quot;. What I mean is that in your integral, s (the side length of the equilateral triangles) should not be the parameter. Instead, the parameter (let&#039;s call it t) should represent the distance from the cross-section to the origin.The line normal to the plane   z  =  1  −  x  −  y is in the direction of the vector ⟨1,1,1⟩, so it&#039;s not hard to check that the intersection of this line with the plane is       1    3        ⟨    1    ,    1    ,    1    ⟩  , which is a distance of       1          3       from the origin. So our t parameter should go from 0 to       1          3      . Then s, the length of the edge of the cross-sectional equilateral triangles, can be expressed in terms of t as   s  =      3        2      t, or as   s  =      6      t.Step 2Now if you compute the integral       ∫    0                  1                  3                                3        4    ⋅  6      t    2    d  t you will get       1    6  , which agrees with your other calculation.

Finding the volume under the plane x+y+z=1.

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