Iterated Integral Question?

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?

Answer 1

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