Five more orders....

Question

Write five other iterated integrals that are equal to the given iterated integral

∫₀⁴ ∫_y^8-y ∫₀^√(4-y) ƒ(x ,y, z) d x d z d y

Answer 1

Firstly draw a figure in the first octant showing the parabolic cylinder y = 4 - x² , and the planes z = y and

y + z = 8 . Notice that both planes are orthogonal to the y z -plane.

A. ∫₀⁴ ∫_y^8-y ∫₀^√(4-y) ƒ(x ,y, z) d x d z d y

B. ∫₀⁴ ∫₀^√(4-y) ∫_y^8-y ƒ(x ,y, z) d z d x d y

C. ∫₀⁴ ∫^z₀ ∫₀^√(4-y) ƒ(x ,y, z) d x d y d z + ∫⁸₄ ∫ ^8-z₀ ∫₀^√(4-y) ƒ(x ,y, z) d x d y d z

E. ∫₀² ∫₀^{4-x^2} ∫_y^8-y ƒ(x ,y, z) d z d y d x

F. ∫₀² ∫₀^{4-x^2}∫₀^zƒ(x,y,z) d yd zd x +∫₀²∫_4-x^2^{4+x^2} ∫₀^{4-x^2}ƒ(x,y,z) d y d zd x+∫₀²∫_4+x^2⁸ ∫₀^8-z ƒ(x,y,z) dydzd x