To evaluate the double integral of $xy + x^2 y^3$ over the region $[0,2] \times [1,3]$, we use iterated integrals. The integral is given by:

[ \iint_{[0,2] \times [1,3]} (xy + x^2 y^3) , dx , dy ]

We can write this as an iterated integral:

$\int_1^3 \left( \int_0^2 (xy + x^2 y^3) \, dx \right) \, dy$

First, we integrate with respect to $x$:

$\int_0^2 (xy + x^2 y^3) \, dx$

This integral can be split into two parts:

$\int_0^2 xy \, dx + \int_0^2 x^2 y^3 \, dx$

We solve these integrals separately:

1. For the first integral:

$\int_0^2 xy \, dx = y \int_0^2 x \, dx = y \left[ \frac{x^2}{2} \right]_0^2 = y \left( \frac{2^2}{2} - \frac{0^2}{2} \right) = y \cdot 2 = 2y$

2. For the second integral:

$\int_0^2 x^2 y^3 \, dx = y^3 \int_0^2 x^2 \, dx = y^3 \left[ \frac{x^3}{3} \right]_0^2 = y^3 \left( \frac{2^3}{3} - \frac{0^3}{3} \right) = y^3 \cdot \frac{8}{3} = \frac{8y^3}{3}$

Adding these two results together:

$\int_0^2 (xy + x^2 y^3) \, dx = 2y + \frac{8y^3}{3}$

Now we integrate this result with respect to $y$:

$\int_1^3 \left( 2y + \frac{8y^3}{3} \right) \, dy$

We split this into two integrals:

$\int_1^3 2y \, dy + \int_1^3 \frac{8y^3}{3} \, dy$

1. For the first integral:

[ \int_1^3 2y , dy = 2 \int_1^3 y , dy = 2 \left[ \frac{y^2}{2} \right]_1^3 = \left[ y^2 \right]_1^3 = 3^2 - 1^2 = 9 - 1 = 8 ]

2. For the second integral:

$\int_1^3 \frac{8y^3}{3} \, dy = \frac{8}{3} \int_1^3 y^3 \, dy = \frac{8}{3} \left[ \frac{y^4}{4} \right]_1^3 = \frac{8}{3} \left( \frac{3^4}{4} - \frac{1^4}{4} \right) = \frac{8}{3} \left( \frac{81}{4} - \frac{1}{4} \right) = \frac{8}{3} \cdot \frac{80}{4} = \frac{8}{3} \cdot 20 = \frac{160}{3}$

Adding these two results together:

$8 + \frac{160}{3} = \frac{24}{3} + \frac{160}{3} = \frac{184}{3}$

So, the value of the double integral is:

[ \iint_{[0,2] \times [1,3]} (xy + x^2 y^3) , dx , dy = \frac{184}{3} ]

Would you like details on any specific part of the solution or have any questions?

Here are 8 related questions you might find interesting:

1. How can the order of integration be changed in a double integral?
2. What are some common techniques for solving double integrals?
3. How do you determine the region of integration for double integrals?
4. What are the applications of double integrals in real-world problems?
5. How do double integrals differ when using polar coordinates?
6. What are some challenges in evaluating double integrals analytically?
7. How does the concept of double integrals extend to triple integrals?
8. What are iterated integrals and how are they useful in evaluating double integrals?

Tip: When integrating composite functions, it’s often useful to break them down into simpler parts and handle each part separately before combining the results.