- #1
yitriana
- 36
- 0
[tex]
\int_{c_1}^{c_2} \int_{g_1 (x)}^{g_2 (x)} f(x,y) dy dx[/tex]
If f(x,y) is function such that it is not easily integrable, if we wanted to switch the bounds of integration so that h1(y) = g1(x) , same for g2(x),
what would be the general way to rewrite the bounds? Would it involve inverse functions?
Let me clarify with a specific (easy) example:
[tex]
\int_{0}^{4} \int_{2}^{\sqrt{y}} e^{x^3} dx dy[/tex],
rewriting sqrt(y) = x as y = x2, and finding intersection points would enable us to rewrite as,
[tex]
\int_{0}^{2} \int_{0}^{x^2} e^{x^3} dy dx[/tex]
and make it possible to integrate.
==
So, how would we rewrite something like
[tex]
\int_{0}^{2} \int_{\sin{y}}^{(y-1)(y-2)(y-3)} e^{x^3} dx dy[/tex]
in solvable terms?
Would we have to find the inverse of sin{y} and the other function as a function of x and make those the bounds for dy?
\int_{c_1}^{c_2} \int_{g_1 (x)}^{g_2 (x)} f(x,y) dy dx[/tex]
If f(x,y) is function such that it is not easily integrable, if we wanted to switch the bounds of integration so that h1(y) = g1(x) , same for g2(x),
what would be the general way to rewrite the bounds? Would it involve inverse functions?
Let me clarify with a specific (easy) example:
[tex]
\int_{0}^{4} \int_{2}^{\sqrt{y}} e^{x^3} dx dy[/tex],
rewriting sqrt(y) = x as y = x2, and finding intersection points would enable us to rewrite as,
[tex]
\int_{0}^{2} \int_{0}^{x^2} e^{x^3} dy dx[/tex]
and make it possible to integrate.
==
So, how would we rewrite something like
[tex]
\int_{0}^{2} \int_{\sin{y}}^{(y-1)(y-2)(y-3)} e^{x^3} dx dy[/tex]
in solvable terms?
Would we have to find the inverse of sin{y} and the other function as a function of x and make those the bounds for dy?