The given expression represents a double integral:

$\int_{0}^{1} \int_{0}^{y} (xy + y^2) \, dx \, dy$

Step 1: Evaluate the inner integral (with respect to $x$)

We first compute the integral of $(xy + y^2)$ with respect to $x$, treating $y$ as a constant.

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

Separate the terms inside the integral:

$= \int_0^y xy \, dx + \int_0^y y^2 \, dx$

Now integrate each term:

1. $\int_0^y xy \, dx = \frac{x^2 y}{2} \bigg|_0^y = \frac{y^3}{2}$
2. $\int_0^y y^2 \, dx = y^2 x \bigg|_0^y = y^3$

Thus, the inner integral becomes:

$\frac{y^3}{2} + y^3 = \frac{3y^3}{2}$

Step 2: Evaluate the outer integral (with respect to $y$)

Now, we compute the outer integral:

$\int_0^1 \frac{3y^3}{2} \, dy$

Factor out $\frac{3}{2}$:

$\frac{3}{2} \int_0^1 y^3 \, dy$

The integral of $y^3$ is:

$\int_0^1 y^3 \, dy = \frac{y^4}{4} \bigg|_0^1 = \frac{1}{4}$

Thus, the outer integral becomes:

$\frac{3}{2} \times \frac{1}{4} = \frac{3}{8}$

Final Answer:

The value of the double integral is $\frac{3}{8}$.

Would you like any further explanation on this process?

Here are 5 related questions for you:

1. What is the geometric interpretation of a double integral?
2. How do you determine the limits of integration for a given region?
3. Can you generalize this technique for integrals with different limits?
4. What happens if you switch the order of integration in a double integral?
5. How would the process change if the integrand involved more complex functions?

Tip: Always check whether the region of integration allows for switching the order of integration to simplify the calculation.