Multivariable Calculus Triple Integration Problem

1. Apr 30, 2014

mathstudent192

1. The problem statement, all variables and given/known data
Express the iterated integral ∫[0,1]∫[0,1-y^2]∫[0,y] f(x,y,z)dzdxdy
a. as a triple integral (i.e., describe the region of integration);
b. as an iterated integral in the order z, y, x;
c. as an iterated integral in the order y, z, x:

3. The attempt at a solution
so would writing (0<=z<=y) (0<=x<=1-y^2) (0<=y<=1) be sufficient for part a.

for part b. i got ∫[0,1] ∫[0,√1-x] ∫[0,y] which i think is correct.

for part c. however i had difficulty in changing the order because when i graph the dz and dy bounds to switch them, dy is in terms of x and I get stuck.

2. Apr 30, 2014

Simon Bridge

(a) A triple integral usually has a triple-integration sign in it.
You are also asked to describe the region of integration - what shape it is? Can you put it in words?
It can help to sketch it out. Though your notes may have a standard approach to how you are supposed to describe the regions.

Understanding this part will help you with the rest.

3. Apr 30, 2014

mathstudent192

would this be an accurate graphing?

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4. Apr 30, 2014

Simon Bridge

Don't forget to label important parts of the sketch ... like where are the bits that are equal to one?
Off context: you appear to have sketched the parabolic surface x=1-y^2 inside 0<z<1 ... is that correct?

Will that give you the integral that you started with?

5. Apr 30, 2014

mathstudent192

ah i see I graphed the z component incorrectly. I'm having trouble visualizing/drawing the 0<z<y part of the graph

6. May 1, 2014

HallsofIvy

$0\le z \le y$ is a plane.