Iterated Integral Question?

∫ (from 0 to 1) [ ∫ from 1 to x) [ ∫ (from 0 to x+y) of y dz ] dy ] dx

also, what would the region of integration look like?

Update:

what happened to the y in the first integral?

1 Answer

Relevance
  • 1 decade ago
    Favorite Answer

    ∫ (from 0 to 1) [ ∫ from 1 to x) [ ∫ (from 0 to x+y) of y dz ] dy ] dx

    --------------------------------

    [ ∫ (from 0 to x+y) of y dz ]

    ....x+y

    z ] ====> x+y

    ....0

    -----------------------

    [ ∫ from 1 to x) [ x+y ] dy

    .................x

    x*y + y^2/2 ] ====> [ x^2 - x ] + [ x^2/2 - (1/2) ] ===> (3/2)x^2 - x - (1/2)

    .................1

    --------------------------

    ∫ (from 0 to 1) [ (3/2)x^2 - x - (1/2) ] dx

    .....................................1

    (3/2)x^3/3 - x^2/2 - (1/2)x ]

    .....................................0

    .....................................1

    (1/2)x^3 - x^2/2 - (1/2)x ] ===> [ (1/2) - 0 ] - [ (1/2) - 0 ] - [ (1/2) - 0 ] = -1/2

    .....................................0

Still have questions? Get your answers by asking now.