Calculating limits of integrals of volumeI have a few problems to do with finding the volume under surfaces. The first ones were fairly simple: h ( x , y ) = x y 2 , 0 ≤ x ≤ 1 , 1 ≤ y ≤ 2 ∴ V = ∫ 1 2 ∫ 0 1 ∫ 0 x y 2 d z d x d y = ∫ 1 2 ∫ 0 1 x y 2 d x d y = 1 2 ∫ 1 2 y 2 d y = 7 6 Step 2However, on later questions the boundaries are given as: h ( x , y ) = e − x − y , 0 ≤ x , y &lt; ∞which to me implies: V = ∫ − ∞ ∞ ∫ 0 ∞ ∫ 0 exp ⁡ ( − x − y ) d z d x d y .However this evaluates to: V = [ e y ] − ∞ ∞ . which is clearly divergent.How should I calculate what the limits are for this volume?

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Calculating limits of integrals of volumeI have a few problems to do with finding the volume under surfaces. The first ones were fairly simple:  h  (  x  ,  y  )  =  x      y    2      ,  0  ≤  x  ≤  1    ,  1  ≤  y  ≤  2  ∴  V  =      ∫    1    2        ∫    0    1        ∫    0          x              y        2              d  z  d  x  d  y  =      ∫    1    2        ∫    0    1    x      y    2    d  x  d  y  =      1    2        ∫    1    2        y    2    d  y  =      7    6  Step 2However, on later questions the boundaries are given as:  h  (  x  ,  y  )  =      e          −      x      −      y          ,  0  ≤  x    ,  y  &amp;lt;  ∞which to me implies:   V  =      ∫          −      ∞        ∞        ∫    0    ∞        ∫    0          exp      ⁡      (      −      x      −      y      )        d  z  d  x  d  y  .However this evaluates to:   V  =            [              e        y            ]              −      ∞        ∞    . which is clearly divergent.How should I calculate what the limits are for this volume?

Harmony Horn · Accepted Answer

Step 1The boundaries   0  ≤  x  ,  y  &amp;lt;  ∞ mean that both x and y can have non-negative values:                    0        ≤        x        &amp;lt;        ∞                            0        ≤        y        &amp;lt;        ∞        .            Step 2So the lower bound in the outer integral (integrating w.r.t y) should be 0.

Find the volume under surfaces. The first ones were fairly simple: h(x,y)=xy^2, 0 leq x leq 1, 1 leq y leq 2

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