You have written a double integral expression, which I assume is meant to evaluate the following:

$\int_0^2 \int_0^{x^2} e^y \, dx \, dy$

Here's how we would approach solving this:

Step 1: Understand the limits and function

The given integral is a double integral, which means we will integrate the function $e^y$ over a two-dimensional region. The limits of integration are:

• For $y$, it goes from 0 to $x^2$.
• For $x$, it goes from 0 to 2.

Thus, the order of integration is: first, with respect to $x$, and then with respect to $y$.

Step 2: Solve the inner integral (with respect to $x$)

We first integrate with respect to $x$. The inner integral is:

$\int_0^2 \left( \int_0^{x^2} e^y \, dy \right) dx$

Since the inner integral is with respect to $y$, and the function $e^y$ is independent of $x$, we can integrate $e^y$:

$\int_0^{x^2} e^y \, dy = \left[ e^y \right]_0^{x^2} = e^{x^2} - e^0 = e^{x^2} - 1$

Step 3: Solve the outer integral (with respect to $x$)

Now we substitute this result back into the outer integral:

$\int_0^2 \left( e^{x^2} - 1 \right) dx$

This is now a single integral in terms of $x$. It involves $e^{x^2}$, which does not have a simple antiderivative in terms of elementary functions. However, we can split the integral as follows:

$\int_0^2 \left( e^{x^2} - 1 \right) dx = \int_0^2 e^{x^2} dx - \int_0^2 1 \, dx$

Step 4: Solve each part separately

• The integral $\int_0^2 1 \, dx$ is straightforward: $\int_0^2 1 \, dx = 2$

• The integral $\int_0^2 e^{x^2} dx$ is a known function that is related to the error function ($\text{erf}(x)$):

$\int_0^2 e^{x^2} dx = \frac{\sqrt{\pi}}{2} \text{erfi}(2)$

Here, $\text{erfi}(x)$ is the imaginary error function, but you would typically approximate it numerically.

Final Answer:

So, the double integral evaluates to:

$\int_0^2 \int_0^{x^2} e^y \, dy \, dx = \left( \frac{\sqrt{\pi}}{2} \text{erfi}(2) - 2 \right)$

Would you like me to continue with numerical approximation or go into further details?

Here are 5 related questions for further exploration:

1. How does changing the limits of integration affect the result of a double integral?
2. What is the error function, and how is it related to integrals involving exponentials?
3. What is the difference between $\text{erf}(x)$ and $\text{erfi}(x)$?
4. Can we solve integrals involving functions like $e^{x^2}$ exactly, or are they usually approximated?
5. How does changing the order of integration impact the result of a double integral?

Tip: When dealing with complex integrals, especially those involving functions like $e^{x^2}$, knowing about special functions (like the error function) can be very helpful.