Question

Evaluate the iterated integrals:
3 1 sqrt(1-z^2)
∫ ∫ ∫ ze^y dxdzdy
0 0 0
Please explain step by step

Answer 1

okay so first step:
isolate the innermost integral (\[\int\limits_{0}^{\sqrt{1-z^2}}ze^ydx\]

Answer 2

since you have dx at the end, you'll be integrating with respect to x. So, you can treat all other variables as constants

Answer 3

are you with me so far?

Answer 4

yes but there is no x so would it be xze^y?

Answer 5

yep! then plug in the limits for x

Answer 6

so i would get (\[\sqrt{1-z ^{2}}ze ^{y}\]

Answer 7

yes

now isolate the second integral

Answer 8

could i sub u = 1- z^2 and get du = -2zdz and dz= du/-2z

Answer 9

and end up getting ∫ uze^y (du/-2z)

Answer 10

which would elnimate z and ive me -1/2∫ u^3/3 (e^y) evaluated from 0 to 1?

Answer 11

good except you'll integrate u^(1/2)*e^y

Answer 12

and substituting the limits youll get from 1 to 0 so multiply the integral by -1 and integrate from 0 to 1

Answer 13

Thanks so much i got it!

Answer 14

you're welcome!