Question

Evaluate double integration 0 to 8 and y^1/3 to 2 (e^x^4) dy dx. why do we need to change the variables in evaluating multiple integrals?

Please answer this question.
Thank you.

Answer 1

See below

Explanation:

Instead of trying to integrate this (not do-able in terms of elementary functions):

• `int_(y = 0)^8 int_(x = y^(1/3))^2 \ e^(x^4) \ dx \ dy`

You can adjust the order [ not change the variables] like this:

• `equiv int_(x = 0)^2 int_(y = 0)^(x^3) \ e^(x^4) \ dy \ dx`

`= int_(x = 0)^2 [y]_(0)^(x^3) \ e^(x^4) \ dx`

`= int_(x = 0)^2 x^3 \ e^(x^4) \ dx`

And that's become trivial