Question

Find the triple integral of (1-z^2)dxdydz of the pyramid. D = [0,1]x[0,1]x[0,1]

Answer 1

\[\int\limits_{0}^{1}\int\limits_{0}^{1}\int\limits_{0}^{1} (1-z^{2})dx dy dz\]

Answer 2

integrate wrt x first, as it is dx dy dz

Answer 3

did you get 2/3?

Answer 4

well, not really. I can't get the book's answer of 3/10

Answer 5

with respect to x, i would get x-xz^2 evaluated from 0 to 1. which brings me back to 1-z^2 dydz

Answer 6

are you sure the question is given as (1-z^2) dx dy dz or do we have (1-z^2) dz dy dx

Answer 7

i am sure

Answer 8

wait a minute, this is pyramid, okay so here is what you need to do

(1-z^2) is bounded by 0 and 1 , now make a project of this 3D figure in 2D ,

Answer 9

the whole question is (1-z^2)dxdydz; W is a pyramid with the top vertex of (0,0,1) and base vertices of (0,0,0), (1,0,0), (0,1,0) and (1,1,0)

Answer 10

and bound of y and x need to be different

Answer 11

oh pellet, change of order integration?

Answer 12

yes

Answer 13

the reason, you dxdydz, so we need either dzdydx or dzdxdy, hope that helps, i am sure youwill have answer 3/10

Answer 14

how would that change the bounds of x and y? would they not still be 0 to 1?

Answer 15

figure not to scale

Answer 16

so z varies from origin to the top most point, and then you can have either projection in XY plane OR CHOOSE OTHER ALTERNATIVEs

Answer 17

ok. thank you