https://file.aiquickdraw.com/mathbot/file/17259802975807fmpchaw.jpg

The expression in the image appears to be:

\[
\int_0^2 \int_0^4 (6xy - 2y) \, dx \, dy
\]

This represents a double integral, where the region of integration is a rectangle with limits \( 0 \leq x \leq 4 \) and \( 0 \leq y \leq 2 \), and the function being integrated is \( 6xy - 2y \).

Let's solve this step by step.

### Step 1: Integrate with respect to \( x \)
The inner integral is:

\[
\int_0^4 (6xy - 2y) \, dx
\]

Treating \( y \) as a constant, we can integrate term by term:

\[
\int_0^4 6xy \, dx = 6y \int_0^4 x \, dx = 6y \left[\frac{x^2}{2}\right]_0^4 = 6y \left[\frac{16}{2}\right] = 48y
\]

\[
\int_0^4 -2y \, dx = -2y \int_0^4 1 \, dx = -2y [x]_0^4 = -2y(4 - 0) = -8y
\]

So the result of the inner integral is:

\[
48y - 8y = 40y
\]

### Step 2: Integrate with respect to \( y \)
Now, we integrate the result from the first step with respect to \( y \):

\[
\int_0^2 40y \, dy = 40 \int_0^2 y \, dy = 40 \left[\frac{y^2}{2}\right]_0^2 = 40 \times \frac{4}{2} = 80
\]

### Final Result:
The value of the double integral is \( 80 \).

Would you like more details on any part of this solution?

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Here are five related questions:
1. How do you determine the limits of integration in a double integral?
2. What is the significance of changing the order of integration in double integrals?
3. Can you evaluate a double integral over non-rectangular regions?
4. What happens if the function within the integral is not separable?
5. How would the result change if the integrand were \( 6xy^2 - 2y \)?

**Tip:** Always check if the function inside the integral allows for simplifications or factorization, as it can make the computation easier.

The query submitted by the user: double integral \( \int_0^2 \int_0^4 (6xy - 2y) \, dx \, dy \)

This solution provides a detailed step-by-step approach to solving the double integral \( \int_0^2 \int_0^4 (6xy - 2y) \, dx \, dy \), covering key concepts in double integration, including integrating with respect to both x and y. Suitable for college-level calculus students.

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