How we solve iterated integrals such as this one?

Question

Here is the problem

$(\textbf{20}\text{ points})$ Calculate the following iterated intergals:

$$\text{a. } \int_0^2\mathrm dx \int_{-1}^1\big(3x^2-(x+y)e^y+xy^3\big)\ \mathrm dy,$$

I can do the next if you explain how to solve this. Thanks!

Answer 1

I personally dislike this notation, and think it's easier to see it written as: $$ \int_0^2\int_{-1}^1(3x^2-xe^y-ye^y+xy^3)\mathop{dy}\mathop{dx} $$

Evaluate the $y$ integral first, considering $x$ as a constant:

\begin{align*} \int_{-1}^1(3x^2-xe^y-ye^y+xy^3) \mathop{dy}&=\left.\left[3x^2y-xe^y-(y-1)e^y+xy^4/4\right]\right|_{y=-1}^{y=1}\\ &=6x^2-\left(e-\frac1e\right)x-\frac2e \end{align*}

Then evaluate the outer integral:

\begin{align*} \int_0^2\left(6x^2-\left(e-\frac1e\right)x-\frac2e\right) \mathop{dx} \end{align*}