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mikau

Member

Registered: 2005-08-22

Posts: 1,504

limits of integration for Triple Integral

Set up the limits of integration for ∫ dV over R where R is the solid formed by the intersection of the parabolic cylinder: z=4-x[sup]2[/sup], the planes z=0, y = x, and y = 0.

Use the following orders of integration:

(1) dx dy dz
(2) dz dx dy

(1)
begining with z, z sweeps from 0 to 4

y then sweeps from 0 to the  y height of the intersection of the cylinder z=4-x[sup]2[/sup] and the plane y=x,  and so we have z=4-y[sup]2[/sup] so y = sqrt(4-z). So y sweeps from 0 to sqrt(4-z)

laslty, x (to my understanding) should sweep from the plane y=x, to the cylinder z=4-x[sup]2[/sup], and so x should sweep from y to sqrt(4-z)

my textbook gives x sweeping from 0 to 2.

(2)
starting with y, y sweeps from 0 to 2

x then sweeps from the plane y=x to 2, that is, from y to 2 (my textbook says from 0 to y)

lastly, z sweeps from 0 to the cylinder z=4-x[sup]2[/sup] so z sweeps from 0 to 4-x[sup]2[/sup]

again, disagreement with the textbook. What am I doing wrong?

Last edited by mikau (2008-11-28 09:05:50)

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