Find the limits of integration ly, uy, lx, ux, lz, uz (some of which will involv

Question

Find the limits of integration ly, uy, lx, ux, lz, uz (some of which will involve variables x, y, z) so that represents the volume of the region in the first octant that is inside the paraboloid and between the planes and

lx =
ux =
lz =
uz =
ly =
uy = Find the limits of integration ly, uy, lx, ux, lz, uz (some of which will involve variables x, y, z) so that dxdzdy represents the volume of the region in the first octant that is inside the paraboloid y = x^2 + 2 z^2 and between the planes y = 8 and y = 10. y = x^2 + 2z^2 y = 8 y = 10.

Explanation / Answer

Only the outer bounds can be numbers only. The inner bounds depend on the function y = x^2 + 2z^2.

x^2 = y - 2z^2

x = +/- sqrt(y - 2z^2)

x bounds are: -sqrt(y - 2z^2) <= x <= sqrt(y - 2z^2)

Since dzdy bounds are outside, you project into the yz plane where the horizontal axis is the y-axis and the vertical axis is the z-axis. In the yz plane, x = 0 so form y = x^2 + 2z^2 we have y = 2z^2.

z^2 = y/2

z = +/-sqrt(y/2)

z bounds are given by: -sqrt(y/2) <= z <= sqrt(y/2)

y bounds are given: 8 <= y <= 10

Lx = -sqrt(y - 2z^2)

Ux = sqrt(y - 2z^2)

Lz = -sqrt(y/2)

Uz = sqrt(y/2)

Ly = 8

Uy = 10

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