Question

Evaluate the iterated integrals Int(Int(Int( ze^y dx dz dy, x goes from 0 to (1-z^2)^1/2, z goes from 0 to 1, y goes from 0 to 3

Answer 1

i know the first step of integration you get \[\int\limits_{0}^{1} \int\limits_{0}^{3}(\sqrt{1-z ^{2}} ze ^{y}) dz dy\] I am just having trouble with the next step. Do i do U substitution?

Answer 2

Well, you're integrating w.r.t z, first, so you treat e\(^y\) as a constant. So you can simply factor it out of the integral. Does that make sense?

Answer 3

Then you can use u-sub, yes. u = 1-x\(^2\), and so forth...

Answer 4

then, bu FToC, you can evaluate from 0<z<3

Answer 5

so \[\int\limits_{0}^{3}-\frac{ 1 }{ 3 }(1-z^2)^{3/2}e ^{y} dy\]

Answer 6

i got it, thanks

Answer 7

Well, you're evaluating from 0<z<3, so you would have constant values.

Answer 8

So, \(\sf \color{red}{z}\) wold not be present any more.

Answer 9

Yeah i entered the equations limits as flip flopped
final answer is \[\frac{ 1 }{ 3 }(e^3-1)\]