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To change the order of integration in this problem, we need to identify the region of integration described by the given bounds.

The integral is:
\[
\int_0^4 \int_{\sqrt{y}}^2 \sqrt{x^3 + 1} \, dx \, dy
\]

### Step 1: Describe the Region of Integration

From the inner integral \(\int_{\sqrt{y}}^2 \, dx\):
- \(x\) goes from \(\sqrt{y}\) to 2.

From the outer integral \(\int_0^4 \, dy\):
- \(y\) goes from 0 to 4.

### Step 2: Find the Boundaries in Terms of \(y\) and \(x\)

1. For \(x\):
   - \(x\) ranges from \(\sqrt{y}\) (which implies \(y = x^2\)) to 2.
   
2. For \(y\):
   - \(y\) ranges from 0 to 4.

When changing the order of integration, we want to find the region in terms of \(x\) as the outer variable.

### Step 3: Determine New Bounds for \(x\) and \(y\)

1. **For \(x\):**
   - Since \(y = x^2\) and \(y\) goes up to 4, \(x\) will range from \(x = 0\) to \(x = 2\) (since \(x = \sqrt{4} = 2\)).

2. **For \(y\):**
   - For a fixed \(x\) in this range, \(y\) goes from \(y = x^2\) (the lower boundary) up to \(y = 4\) (the upper boundary).

### Step 4: Rewrite the Integral

The new integral, with the order of integration changed, is:
\[
\int_0^2 \int_{x^2}^4 \sqrt{x^3 + 1} \, dy \, dx
\]

### Final Answer

\[
\int_0^2 \int_{x^2}^4 \sqrt{x^3 + 1} \, dy \, dx
\]

Would you like further details or explanations for any of these steps?

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Here are five related questions to deepen your understanding:

1. How does changing the order of integration affect the calculation of the integral?
2. What are the typical methods for visualizing the region of integration in double integrals?
3. How would the bounds change if the integrand or the region had different limits?
4. Why is it useful to change the order of integration in certain double integrals?
5. How would we handle a similar problem if the region of integration was more complex (e.g., triangular or circular)?

**Tip:** When changing the order of integration, sketching the region can be very helpful in determining the new bounds accurately.

Change the order of integration for the double integral \( \int_0^4 \int_{\sqrt{y}}^2 \sqrt{x^3 + 1} \, dx \, dy \) and express the answer in the form \( \int_a^b \int_{g_1(x)}^{g_2(x)} f(x, y) \, dy \, dx \).

Explore how to change the order of integration for the double integral \( \int_0^4 \int_{\sqrt{y}}^2 \sqrt{x^3 + 1} \, dx \, dy \). This solution explains how to identify the region of integration and adjust the bounds to achieve the desired form \( \int_a^b \int_{g_1(x)}^{g_2(x)} f(x, y) \, dy \, dx \).

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