The image shows a question about solving a double integral with several answer choices. I will transcribe and analyze the mathematical content of the problem.

The question seems to be: $\int_0^2 \int_0^x (x + y)\, dy \, dx$

This is a double integral where we are integrating the function $f(x, y) = x + y$ with respect to $y$, then with respect to $x$, and the limits are from 0 to $x$ for $y$, and from 0 to 2 for $x$.

Let’s solve this step by step.

Step 1: Inner integral with respect to $y$

We first integrate $x + y$ with respect to $y$, treating $x$ as a constant: $\int_0^x (x + y) \, dy = \int_0^x x \, dy + \int_0^x y \, dy$ The first part is: $\int_0^x x \, dy = x \cdot y \Big|_0^x = x^2$ The second part is: $\int_0^x y \, dy = \frac{y^2}{2} \Big|_0^x = \frac{x^2}{2}$ Adding both parts together: $x^2 + \frac{x^2}{2} = \frac{3x^2}{2}$

Step 2: Outer integral with respect to $x$

Now we integrate the result $\frac{3x^2}{2}$ with respect to $x$: $\int_0^2 \frac{3x^2}{2} \, dx = \frac{3}{2} \int_0^2 x^2 \, dx = \frac{3}{2} \cdot \frac{x^3}{3} \Big|_0^2$ $= \frac{3}{2} \cdot \frac{8}{3} = 4$

So, the value of the double integral is $4$.

Final Answer:

The correct answer is $4$.

Do you have any questions, or would you like more details on any of the steps?

5 Related Questions:

1. How do you determine the limits of integration for double integrals?
2. What is the geometric interpretation of a double integral?
3. How do you solve a double integral when the limits are functions of one variable?
4. What are the common mistakes made when evaluating double integrals?
5. Can this problem be solved using polar coordinates?

Tip:

When solving double integrals, always make sure to integrate in the correct order (inside-out), and carefully check the limits of integration for each variable.