The image you uploaded appears to show a problem involving the double integral of a function $f(x, y) = x y^2$ over a rectangle $R = [0, 3] \times [1, 3]$. Let's break down and solve the questions presented in the image:

1. Estimate the double integral using a Riemann sum with $m = 2$, $n = 2$, and upper-right corners for the rectangles.

We will approximate the double integral using a Riemann sum, where we divide the rectangle $R$ into smaller subrectangles. For $m = 2$ and $n = 2$, this divides the interval $[0, 3]$ into two subintervals of length 1.5 each, and the interval $[1, 3]$ into two subintervals of length 1 each.

The upper-right corner means we will evaluate the function $f(x, y)$ at the top-right corner of each subrectangle.

Let's calculate the sum:

• The subintervals in $x$ are $[0, 1.5]$ and $[1.5, 3]$
• The subintervals in $y$ are $[1, 2]$ and $[2, 3]$

We evaluate $f(x, y) = x y^2$ at the points $(1.5, 2), (1.5, 3), (3, 2), (3, 3)$.

2. Estimate the double integral using a Riemann sum with $m = 2$, $n = 2$, and the Midpoint Rule.

For the Midpoint Rule, we need to evaluate $f(x, y)$ at the midpoints of each subrectangle. The midpoints of the subrectangles are:

• $\left( 0.75, 1.5 \right), \left( 0.75, 2.5 \right), \left( 2.25, 1.5 \right), \left( 2.25, 2.5 \right)$
• Then evaluate the function at these points and sum.

3. Find the exact value of the double integral.

To find the exact value, we need to compute the integral of $f(x, y) = x y^2$ over the region $R = [0, 3] \times [1, 3]$. We integrate $f(x, y)$ first with respect to $x$ and then with respect to $y$.

$\int_{1}^{3} \left( \int_{0}^{3} x y^2 \, dx \right) dy$

4. Find the average value of $f(x, y)$ over $R$.

The average value of a function over a region $R$ is given by:

$\text{Average value} = \frac{1}{\text{Area of } R} \int \int_{R} f(x, y) \, dA$

We will compute this using the exact integral result from question 3.

Let me know if you'd like the detailed solutions to any of these steps or have any further questions!