Double Integrals and finding the volume of a poolI am trying to find the volume of a pool and Im a little rusty on the calculus. How would I go about finding the volume of pool that is Length 10ft by Width 8ft? The depth goes from 3 to 6 feet with a slope of 3ft/10ft.

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Andre Ferguson · Accepted Answer

Step 1The pool is a cuboid of length 10 ft breadth 8 ft and height 3 ft placed over an inverted wedge of dimensions base length 10 ft, base breadth 8 ft and height 3 ft.Thus, volume of the pool is the volume of the cuboid plus the volume of the inverted wedge i.e,      V          pool        =      V          cuboid        +      V          inverted wedge        (  8  ×  10  ×  3  )  +  (      1    2    ×  8  ×  10  ×  3  )  =  360       ft    3  I guess if you really wanted it in terms of integration then we can derive these volumes via integration.Step 2The area of the cross section of the wedge is given by      ∫          0              8            3    8    y  d  yAnd the volume will be the integration of this cross section over the base length i.e.      V          inverted wedge        =      ∫          0              10            (          ∫              0                    8                    3      8        y    d    y    )    d  xYou can do the same for a cuboid. You should get      V          cuboid        =      ∫          0              10            (          ∫              0                    8              3    d    y    )    d  x

How would I go about finding the volume of pool that is Length 10ft by Width 8ft?

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