To sketch the region of integration for the given double integrals, let’s analyze the bounds of each integral.

First Integral

$\int_{-4}^{0} \int_{0}^{8+2x} f(x,y) \, dy \, dx$

1. Outer Integral: $x$ goes from $-4$ to $0$.
2. Inner Integral: $y$ goes from $0$ to $8 + 2x$.

This suggests that $y$ is bounded by $y = 0$ (the x-axis) and $y = 8 + 2x$. To understand this bound:

• Rewrite $y = 8 + 2x$ as $x = \frac{y - 8}{2}$, which is a line with a slope of $\frac{1}{2}$ and y-intercept at $8$.

So, for the first integral, the region lies between $x = -4$ and $x = 0$ and is bounded above by the line $y = 8 + 2x$ and below by $y = 0$.

Second Integral

$\int_{0}^{8} \int_{0}^{4 - \frac{y}{2}} f(x,y) \, dx \, dy$

1. Outer Integral: $y$ goes from $0$ to $8$.
2. Inner Integral: $x$ goes from $0$ to $4 - \frac{y}{2}$.

This suggests that $x$ is bounded by $x = 0$ (the y-axis) and $x = 4 - \frac{y}{2}$. To understand this bound:

• Rewrite $x = 4 - \frac{y}{2}$ as $y = 8 - 2x$, which is also a line but with a slope of $-2$ and y-intercept at $8$.

For the second integral, the region lies between $y = 0$ and $y = 8$ and is bounded on the right by the line $x = 4 - \frac{y}{2}$.

Combined Region

The region of integration is the intersection of these two regions. To sketch it:

1. Draw the line $y = 8 + 2x$, which passes through points $(-4, 0)$ and $(0, 8)$.
2. Draw the line $y = 8 - 2x$, which passes through points $(0, 8)$ and $(4, 0)$.
3. The region of interest is the area where these bounds intersect.

This region will look like a quadrilateral bounded by the lines:

• $y = 0$ (x-axis),
• $y = 8 + 2x$,
• $y = 8 - 2x$,
• and vertical bounds $x = -4$ to $x = 4$.

Would you like additional details on any step, or a further breakdown of how these integrals define the region?

Here are five related questions for further exploration:

1. How would the region change if the bounds were altered in the integrals?
2. What would the graph look like if one integral had reversed limits?
3. Can you verify the intersection points of the lines defining the region?
4. How would changing the function $f(x, y)$ impact the evaluation of these integrals?
5. Could this region be represented in polar coordinates, and if so, how?

Tip: When analyzing regions of integration, always start by understanding each bound separately before combining them.