I have this integral:
$$\int_0^1\int_0^{\arccos(y)}x^2\sec(x)\,dx\,dy$$
When I reverse the order of integration, what will be its new limits? I think the new limits will be as following:
$0 \le y \le \cos(x)$ and $0 \le x \le 1$
So,
- Can someone tell me if I am correct
- Is it possible if you can show shape of the region of integration and the orientation of the infinitesimal area elements for both the original and the reversed order of integration.
You are not correct... the graphic of $x =\arccos y$ crosses the $x$-axis at $x =\pi/2$. So, the proper integration limits would be: $$ \int_0^{\pi/2} \int_{0}^{\cos x} \frac{x^2}{\cos x} dy dx $$
You should keep in mind that this is an improper integral.