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Let's analyze the given problem. We are asked to determine the limits of integration for the double integral:

\[
I = \int \int_D f(x, y) \, dx \, dy
\]
where the region \( D \) is bounded by the curves \( y = 2 \) and \( y^2 = x \), and the y-axis (i.e., \( x = 0 \)).

### Understanding the region \( D \):
1. The curve \( y^2 = x \) is a parabola that opens to the right. Rewriting this as \( x = y^2 \), it tells us that for a given \( y \)-value, \( x \) ranges from 0 to \( y^2 \).
2. The line \( y = 2 \) is a horizontal line that intersects the parabola.

### Setting up the limits of integration:
- The region \( D \) is bounded on the left by \( x = 0 \) (the y-axis) and on the right by \( x = y^2 \).
- The \( y \)-values range from \( y = 0 \) to \( y = 2 \).

Thus, the correct setup for the double integral in terms of \( dx \, dy \) would be:
\[
I = \int_0^2 \int_0^{y^2} f(x, y) \, dx \, dy
\]

### Comparing the options:
- Option **B** matches this setup exactly.

Thus, the correct answer is **B**.

Would you like a further breakdown or details on how to approach double integrals?

### 5 Relative Questions:
1. How can you verify the boundaries of a double integral from a given region?
2. What is the significance of the order of integration in double integrals?
3. How do you determine whether to use Cartesian or polar coordinates for a double integral?
4. How would the integral change if the region were bounded by different curves?
5. Can you solve a similar integral but with different functions for the boundaries?

### Tip:
Always sketch the region of integration for double integrals. This helps visualize the limits and ensures the setup is correct.

Xác định cận lấy tích phân I = ∫∫_D f(x,y) dx dy với miền lấy tích phân D được giới hạn bởi các đường OY, y=2, y^2 = x.

This page explains how to determine the integration limits for a double integral given the region bound by y=2, y^2=x, and OY. It includes detailed explanations and steps for setting up the integral correctly.

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