Estimate the volume of the solid that lies below the surface z = xy and above the following rectangle.

R =

(x, y)|8 ≤ x ≤ 14, 4 ≤ y ≤ 8

Use a Riemann sum with m = 3, n = 2, and take the sample point to be the upper right corner of each square.

(b) Use the Midpoint Rule to estimate the volume of the solid in part (a)

thanks so much

To estimate the volume of the solid using a Riemann sum, we will first divide the given rectangle R into 6 smaller rectangles (m=3 along the x-axis, n=2 along the y-axis). The dimensions of each small rectangle along the x-axis will be Δx = (14-8)/3 = 2 and along the y-axis will be Δy = (8-4)/2 = 2.

Next, we will evaluate the function z = xy at the upper right corner (sample point) of each small rectangle, and then multiply each value by the area of the rectangle (ΔxΔy = 2*2 = 4).
For the sample points, we only pick upper right corner for each small rectangle:

1. (x1, y1) = (10, 6)
2. (x2, y2) = (12, 6)
3. (x3, y3) = (14, 6)
4. (x4, y4) = (10, 8)
5. (x5, y5) = (12, 8)
6. (x6, y6) = (14, 8)

Now, let's compute the volume of each small rectangular box:

1. V1 = f(x1, y1) * ΔxΔy = (10 * 6) * 4 = 240
2. V2 = f(x2, y2) * ΔxΔy = (12 * 6) * 4 = 288
3. V3 = f(x3, y3) * ΔxΔy = (14 * 6) * 4 = 336
4. V4 = f(x4, y4) * ΔxΔy = (10 * 8) * 4 = 320
5. V5 = f(x5, y5) * ΔxΔy = (12 * 8) * 4 = 384
6. V6 = f(x6, y6) * ΔxΔy = (14 * 8) * 4 = 448

Now, adding these volumes up, we get the estimated volume of the solid as:

V = V1 + V2 + V3 + V4 + V5 + V6 = 240 + 288 + 336 + 320 + 384 + 448 = 2016

So, the estimated volume of the solid using a Riemann sum is 2016 cubic units.

(b) Using the Midpoint Rule, we will find the midpoints of each small rectangle and evaluate the function z = xy at these midpoints. Then, we will multiply the value by the area of the rectangle (ΔxΔy = 4).

Now, let's find the midpoints for each small rectangle:

1. (x1, y1) = (9, 5)
2. (x2, y2) = (11, 5)
3. (x3, y3) = (13, 5)
4. (x4, y4) = (9, 7)
5. (x5, y5) = (11, 7)
6. (x6, y6) = (13, 7)

Now, let's compute the volume of each small rectangular box using the midpoints:

1. V1 = f(x1, y1) * ΔxΔy = (9 * 5) * 4 = 180
2. V2 = f(x2, y2) * ΔxΔy = (11 * 5) * 4 = 220
3. V3 = f(x3, y3) * ΔxΔy = (13 * 5) * 4 = 260
4. V4 = f(x4, y4) * ΔxΔy = (9 * 7) * 4 = 252
5. V5 = f(x5, y5) * ΔxΔy = (11 * 7) * 4 = 308
6. V6 = f(x6, y6) * ΔxΔy = (13 * 7) * 4 = 364

Now, adding these volumes up, we get the estimated volume of the solid using the Midpoint Rule as:

V = V1 + V2 + V3 + V4 + V5 + V6 = 180 + 220 + 260 + 252 + 308 + 364 = 1584

So, the estimated volume of the solid using the Midpoint Rule is 1584 cubic units.

To estimate the volume of the solid using the Midpoint Rule, we need to divide the rectangular region into smaller subrectangles and find the volume of each subrectangle. Here's how you can do that:

Step 1: Divide the x-interval [8, 14] into three equal subintervals since m = 3. This gives us the following x-values for the vertical lines of division: 8, 10, 12, 14.

Step 2: Divide the y-interval [4, 8] into two equal subintervals since n = 2. This gives us the following y-values for the horizontal lines of division: 4, 6, 8.

Step 3: Using the Midpoint Rule, the sample point for each subrectangle is the upper right corner. So, for each subrectangle, we take the midpoint of the x-interval as the x-coordinate and the upper bound of the y-interval as the y-coordinate.

Step 4: Calculate the volume of each subrectangle using the formula V = Δx * Δy * Δz, where Δx is the width of the subrectangle, Δy is the height of the subrectangle, and Δz is the difference between the function values at the sample point and the y-coordinate of the rectangle.

Step 5: Sum up the volumes of all the subrectangles to get the estimated volume of the solid.

Let's calculate it step-by-step:

Subrectangle 1: (8, 4) to (10, 6)
- Δx = 10 - 8 = 2
- Δy = 6 - 4 = 2
- Sample point: (9, 6)
- Δz = f(9, 6) - 6 = (9 * 6) - 6 = 48 - 6 = 42
- Volume = Δx * Δy * Δz = 2 * 2 * 42 = 168

Subrectangle 2: (10, 4) to (12, 6)
- Δx = 12 - 10 = 2
- Δy = 6 - 4 = 2
- Sample point: (11, 6)
- Δz = f(11, 6) - 6 = (11 * 6) - 6 = 66 - 6 = 60
- Volume = Δx * Δy * Δz = 2 * 2 * 60 = 240

Subrectangle 3: (12, 4) to (14, 6)
- Δx = 14 - 12 = 2
- Δy = 6 - 4 = 2
- Sample point: (13, 6)
- Δz = f(13, 6) - 6 = (13 * 6) - 6 = 78 - 6 = 72
- Volume = Δx * Δy * Δz = 2 * 2 * 72 = 288

Subrectangle 4: (8, 6) to (10, 8)
- Δx = 10 - 8 = 2
- Δy = 8 - 6 = 2
- Sample point: (9, 8)
- Δz = f(9, 8) - 8 = (9 * 8) - 8 = 72 - 8 = 64
- Volume = Δx * Δy * Δz = 2 * 2 * 64 = 256

Subrectangle 5: (10, 6) to (12, 8)
- Δx = 12 - 10 = 2
- Δy = 8 - 6 = 2
- Sample point: (11, 8)
- Δz = f(11, 8) - 8 = (11 * 8) - 8 = 88 - 8 = 80
- Volume = Δx * Δy * Δz = 2 * 2 * 80 = 320

Subrectangle 6: (12, 6) to (14, 8)
- Δx = 14 - 12 = 2
- Δy = 8 - 6 = 2
- Sample point: (13, 8)
- Δz = f(13, 8) - 8 = (13 * 8) - 8 = 104 - 8 = 96
- Volume = Δx * Δy * Δz = 2 * 2 * 96 = 384

Finally, sum up the volumes of all the subrectangles:
Estimated volume = 168 + 240 + 288 + 256 + 320 + 384 = 1656 cubic units.

Therefore, the estimated volume of the solid using the Midpoint Rule is 1656 cubic units.

To estimate the volume of the solid using the Midpoint Rule, we need to divide the given rectangle R into smaller squares and find the approximate volume of each square.

The given rectangle R is defined as: R = {(x, y) | 8 ≤ x ≤ 14, 4 ≤ y ≤ 8}

We will divide this rectangle into smaller squares by setting m = 3 and n = 2. This means there will be 3 squares in the x-direction and 2 squares in the y-direction.

First, let's calculate the width and height of each square:
Delta x = (14 - 8) / 3 = 2
Delta y = (8 - 4) / 2 = 2

Now, let's calculate the coordinates of the midpoint of each square in the x and y directions:
x-midpoint = x-coordinate of the left side + (Delta x / 2)
y-midpoint = y-coordinate of the bottom side + (Delta y / 2)

x-midpoints: 9, 11, 13
y-midpoints: 5, 7

Now, we need to evaluate the function z = xy at each midpoint and multiply it by the area of each square.

For the square with the upper right corner at (9, 5):
z(9, 5) = 9 * 5 = 45

For the square with the upper right corner at (9, 7):
z(9, 7) = 9 * 7 = 63

For the square with the upper right corner at (11, 5):
z(11, 5) = 11 * 5 = 55

For the square with the upper right corner at (11, 7):
z(11, 7) = 11 * 7 = 77

For the square with the upper right corner at (13, 5):
z(13, 5) = 13 * 5 = 65

For the square with the upper right corner at (13, 7):
z(13, 7) = 13 * 7 = 91

Now, we multiply each result by the area of each square, which is Delta x * Delta y = 2 * 2 = 4.

Volume of the solid = (45 * 4) + (63 * 4) + (55 * 4) + (77 * 4) + (65 * 4) + (91 * 4) = 180 + 252 + 220 + 308 + 260 + 364 = 1584

Therefore, the approximate volume of the solid using the Midpoint Rule is 1584 cubic units.