How do I evaluate this double integral by changing the order of integration?

Question

One of my past papers has the following question (without a solution).

Evaluate the integral by changing the order of integration.

$\int_0^1dx$$\int_{x^{1/a}}^1e^{y^{a+1}}dy$

where $a$ is a constant and $a \neq -1, 0$.

I would assume changing the order of integration would make it easier to do but I can't see this.

Answer 1

Draw on a graph and convince yourself that the region bounded by $x = 0 , x = 1$ and $y$ varying from $x^{1/a}$ to $1$ is same as the region enclosed by $y =0, y = 1$, and $x$ varying from $0$ to $y^a$, and change the integral accordingly

Answer 2

From $x^{\frac1a}\leq y\leq1$ we have $x\leq y^a\leq1$ and also $0\leq y\leq1$ then the integral is $$\int_0^1\,\mathrm{dy}\int_0^{y^a}e^{y^{a+1}}\,\mathrm{dx}=\int_0^1\,y^a\,e^{y^{a+1}}\,\mathrm{dy}=\color{blue}{\dfrac{1}{a+1}(e-1)}$$