<s>[INST] Evaluate $\int_0^\frac{1}{16}\int_{y^{\frac{1}{4}}}^\frac{1}{4} \cos{\left(16\pi x^5\right)}dx\,dy$ $\displaystyle\int_0^\frac{1}{16}\int_{y^{\frac{1}{4}}}^\frac{1}{4} \cos{\left(16\pi x^5\right)dx\,dy}$
My attempt:
Reverse the order of integration

Then the integral will be:
$$\displaystyle\int_{x^4}^\frac{1}{16}\int_{0}^\frac{1}{4} \cos{\left(16\pi x^5\right)dx\,dy}$$
Using wolfram, this is what I found:
$$0.0154$$
But the correct answer is $-0.0085$. Where did I go wrong?
 [/INST] The region of integration has two wings. Thus in reversing the order of integration you are going to have two integrals namely  $$\displaystyle\int_0^\frac{1}{16}\int_{y^{\frac{1}{4}}}^\frac{1}{4} \cos{\left(16\pi x^5\right)dx\,dy}=\displaystyle\int_{0}^\frac{1}{4}\int_{0}^{x^4} \cos{\left(16\pi x^5\right)dy\,dx} -$$
$$\displaystyle\int_{\frac {1}{4}}^\frac{1}{2}\int_{x^4}^\frac{1}{16} \cos{\left(16\pi x^5\right)dy\,dx}$$
</s>