Finding volumes - when to use double integrals and triple integrals?This is not a technical question at all, but I'm quite confused about what should I use to compute volumes in R 3 with integration.I've read somewhere that a double integral gets the volume swapping across the x and y axis while a triple integral just integrate the whole thing at once, how accurate is this?. Can a volume expressed by a double integral be expressed by a triple integral?, and a triple can always be expressed by a double?

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Finding volumes - when to use double integrals and triple integrals?This is not a technical question at all, but I&#039;m quite confused about what should I use to compute volumes in             R        3   with integration.I&#039;ve read somewhere that a double integral gets the volume swapping across the x and y axis while a triple integral just integrate the whole thing at once, how accurate is this?. Can a volume expressed by a double integral be expressed by a triple integral?, and a triple can always be expressed by a double?

Miguel Shah · Accepted Answer

Step 1You can use both double and triple integrals when calculating a volume. Let me explain you using an example for calculating an area, same applies to volume.Say you are looking to find the area under a curve   f  (  x  )  &amp;gt;  0 over the domain of integration. You must have learn that:  A  =      ∫    b    a    f  (  x  )  d  xWhat you are doing is basically summing infinitely many stripes of length f(x) and base length dx.Step 2Now observe that the following are the same:      ∫    b    a        ∫    0          f      (      x      )        d  t  d  x  =      ∫    b    a    f  (  x  )  d  x  =  ASo to calculate the area I can use both single and double integral. The second one is simply a shortcut. The first integral sums infinitely many little square of dimension   d  t  ×  d  x within the specified bounds for t and x.Similarly to find volumes:  ∫  ∫      ∫    0          f      (      x      ,      y      )        d  t  d  x  d  y  =  ∫  ∫  f  (  x  ,  y  )  d  x  d  yThe only difference is that the triple integral is a more basic approach in the sense that you really do it small cube by small cube. The double integral is a shortcut.

Ivan Buckley · Accepted Answer

Step 1Consider a rectangle with sides a, b, c. Its volume is       ∫    0    a        ∫    0    b        ∫    0    c    1 or       ∫    0    a        ∫    0    b    c or       ∫    0    a    b  c. Same logic can be applied to any other shape. Volume is a single integral of area of cross section or a double integral of height.Step 2Note that this is not the only way to split shapes though. For example, you could find the volume of a sphere by integrating over surfaces of all spheres inside it with the same center.

Double integral gets the volume swapping across the x and y axis while a triple integral just integrate the whole thing at once, how accurate is this? Can a volume expressed by a double integral be expressed by a triple integral? And can a triple always be expressed by a double?

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