https://file.aiquickdraw.com/mathbot/file/17255376976872gc85wnb.jpg

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.

Learn how to solve the double integral ∫0^2 ∫0^x (x + y) dy dx with a detailed step-by-step solution. This calculus problem involves calculating the integral with respect to y first and then x. Suitable for undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation