Multivariable Calculus Triple Integration Problem

[mathstudent192]

Homework Statement

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:

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.

 

[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.

 

[mathstudent192]

would this be an accurate graphing?

 

[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?

 

[mathstudent192]

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

 

[HallsofIvy]

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