To solve this problem, we need to reverse the order of integration for the given double integral:

$\int_{-1}^{3} \int_{y^2 + 2}^{2y + 5} f(x, y) \, dx \, dy$

Step 1: Understand the region of integration

The original limits of integration are:

• For $y$: $-1 \leq y \leq 3$
• For $x$: $y^2 + 2 \leq x \leq 2y + 5$

This represents a region in the $xy$-plane where $x$ varies between $y^2 + 2$ and $2y + 5$, and $y$ varies between $-1$ and $3$. To reverse the order of integration, we need to:

1. Identify the region of integration in the $xy$-plane.
2. Express this region with $x$ as the outer integral and $y$ as the inner integral.

Step 2: Analyze the boundary curves

The bounds for $x$ are given by: $x = y^2 + 2 \quad \text{(parabolic curve)}$ $x = 2y + 5 \quad \text{(linear curve)}$

We also know that $y$ ranges from $-1$ to $3$.

To find the points of intersection between these two curves, solve: $y^2 + 2 = 2y + 5$ Simplify: $y^2 - 2y - 3 = 0$ Factorize: $(y - 3)(y + 1) = 0$ Thus, the points of intersection are $y = -1$ and $y = 3$.

So, the region of integration is bounded by:

1. $x = y^2 + 2$ (below the curve).
2. $x = 2y + 5$ (above the curve).
3. From $y = -1$ to $y = 3$.

Step 3: Reverse the order of integration

Now, we switch the order of integration, where $x$ becomes the outer integral. The new limits are:

• For $x$: The minimum value of $x$ is at $y^2 + 2$ when $y = -1$, which is $x = (-1)^2 + 2 = 3$. The maximum value of $x$ is at $2y + 5$ when $y = 3$, which is $x = 2(3) + 5 = 11$. So, $x$ varies from $3$ to $11$.
• For $y$: For a fixed $x$, $y$ varies between the curves $x = y^2 + 2$ and $x = 2y + 5$. Solving for $y$:
  • From $x = y^2 + 2$: $y = \pm\sqrt{x - 2}$
  • From $x = 2y + 5$: $y = \frac{x - 5}{2}$

Thus, the new integral becomes: $\int_{3}^{11} \int_{\sqrt{x-2}}^{\frac{x-5}{2}} f(x, y) \, dy \, dx$

Final Answer:

The equivalent integral with the order of integration reversed is: $\int_{3}^{11} \int_{\sqrt{x-2}}^{\frac{x-5}{2}} f(x, y) \, dy \, dx$

Would you like a detailed explanation of any step, or clarification on the bounds?

5 Related Questions:

1. How do you determine the intersection points of curves in general?
2. What are the general steps to reverse the order of integration in double integrals?
3. Why is it important to carefully analyze the region of integration when reversing limits?
4. How do parabolic and linear boundaries affect the integration setup?
5. Can this process be extended to triple integrals?

Tip:

When reversing the order of integration, always sketch the region to avoid mistakes with the bounds!