Evaluate ffxy ddy over the region bounded by x axis, ordinate x=2a and the curve x^2=4ay.

To evaluate the double integral \( \iint_R f(x, y) \, dy \, dx \) over the region bounded by the x-axis, the ordinate \( x = 2a \), and the curve \( x^2 = 4ay \), we must first understand the region \( R \) in the xy-plane.

### Step 1: Analyze the boundaries
- The curve \( x^2 = 4ay \) is a parabola that opens upwards with its vertex at the origin \((0, 0)\).
- The x-axis is \( y = 0 \).
- The vertical line is \( x = 2a \).

### Step 2: Express the region of integration
The region \( R \) is bounded by:
- The x-axis \( y = 0 \)
- The parabola \( y = \frac{x^2}{4a} \)
- The vertical line \( x = 2a \)

### Step 3: Determine the limits of integration
For a fixed value of \( x \) ranging from \( 0 \) to \( 2a \), \( y \) will range from the x-axis \( y = 0 \) to the parabola \( y = \frac{x^2}{4a} \).

Thus, the integral is set up as follows:
\[
\int_{x=0}^{2a} \int_{y=0}^{\frac{x^2}{4a}} f(x, y) \, dy \, dx
\]

### Final Answer
The double integral over the given region is:
\[
\iint_R f(x, y) \, dy \, dx = \int_{x=0}^{2a} \int_{y=0}^{\frac{x^2}{4a}} f(x, y) \, dy \, dx
\]

If you have a specific function \( f(x, y) \) in mind, we can proceed to evaluate the integral by substituting it in.

Would you like to evaluate this integral for a specific function, or do you have any other questions?  

Here are 5 related questions to expand on this:

1. How would the limits of integration change if the region were bounded by \( y = 0 \), \( y = x^2/4a \), and \( x = -2a \)?
2. What if the function \( f(x, y) \) were a simple polynomial like \( f(x, y) = xy \)? How would you solve the integral?
3. How would you approach the problem if the curve were \( x^2 = 4ay \) but the region were bounded by \( x = 3a \) instead of \( x = 2a \)?
4. Can this integral be evaluated using polar coordinates?
5. What is the geometric interpretation of this double integral?

**Tip:** Always sketch the region of integration first when solving double integrals to ensure the correct limits are chosen.

This problem involves evaluating a double integral over a region defined by the x-axis, a vertical line x = 2a, and a parabolic curve x^2 = 4ay. Learn how to determine the limits of integration and set up the integral step-by-step. Suitable for undergraduate students studying Calculus II or III.

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