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The problem asks to compute the double integral:

\[
\int\int_D (y + y^3) \, dx \, dy
\]

where the region \( D \) is in the second quadrant, bounded by the coordinate axes and the curve \( y = \sqrt{x} + 1 \).

### Step-by-step approach:

1. **Understand the region \( D \):**
   - The curve \( y = \sqrt{x} + 1 \) is given. This can be rewritten as \( x = (y - 1)^2 \) to express \( x \) as a function of \( y \).
   - The region lies in the **second quadrant**. In this quadrant, \( x \leq 0 \) and \( y \geq 0 \). This means we consider the reflection of the curve into the second quadrant, which gives us the bounds for \( x \) and \( y \).
   - Thus, the region is bounded between the vertical line \( x = 0 \) (the y-axis), the curve \( y = \sqrt{x} + 1 \) mirrored into the second quadrant, and the x-axis.

2. **Set up the limits of integration:**
   - For \( y \), it starts from \( y = 1 \) and goes up to some upper limit, which is determined by the points where the curve intersects the boundary.
   - For \( x \), we will use \( x = -(y-1)^2 \) since we are in the second quadrant (negative \( x \)-values).

3. **Rewrite the double integral:**
   \[
   \int_1^a \int_{-(y-1)^2}^0 (y + y^3) \, dx \, dy
   \]
   The inner integral is with respect to \( x \), where \( y + y^3 \) is constant for each value of \( y \).

4. **Evaluate the inner integral:**
   \[
   \int_{-(y-1)^2}^0 (y + y^3) \, dx = (y + y^3) \cdot \left[ 0 - (-(y-1)^2) \right] = (y + y^3) \cdot (y-1)^2
   \]

5. **Substitute and evaluate the outer integral:**
   Now we integrate the result with respect to \( y \) from 1 to the upper bound \( a \), which will need to be defined based on the geometry of the curve.

Would you like me to proceed with these calculations and clarify the final bounds, or should I adjust any steps or explanations? Feel free to ask for more details.

Here are 5 questions to expand this topic:

1. What is the significance of the second quadrant for this problem?
2. How do you determine the bounds of integration for a double integral?
3. Can this integral be evaluated using polar coordinates?
4. How does the equation \( y = \sqrt{x} + 1 \) affect the region of integration?
5. What are some methods to simplify integrals involving cubic terms?

**Tip**: When setting up double integrals, always visualize the region of integration carefully to ensure proper bounds.

Beräkna dubbelintegralen där D är området i andra kvadranten som begränsas av koordinataxlarna och kurvan y = √x + 1.

This problem involves computing a double integral ∫∫_D (y + y^3) dx dy where the region D lies in the second quadrant and is bounded by the axes and the curve y = √x + 1. The solution includes determining the integration boundaries and performing the necessary calculations.

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