The volume of the solid that lies below the surface z = xy and above the following rectangle. R = (x, y) | 2 ≤ x ≤ 8, 6 ≤ y ≤ 10 is 1296
Using the Midpoint Rule, the volume of the solid is 960
Let's consider the surface z = xy; where;
- [tex]\mathbf{R = \{(x,y)\Big |2\leq x\leq 8 \ , \ 6 \leq y \leq 10\}}[/tex]
Using the Reinmann sum to determine the volume of the solid by taking m = 3 and n = 2
The area (ΔA) of the square at the upper right end corner from the graph attached below with the side length 2 is:
= 2²
= 4
Now, the volume of the solid according to the Riemann sum can be computed by using the formula:
[tex]\mathbf{Volume (V) = \sum \limits^{m}_{i=1} \ \sum \limits^{n}_{j=1} \ f(x_1,y_j) \Delta A}[/tex]
where;
Following the midpoints from the graph;
[tex]\mathbf{Volume (V) =\Big( f(4,8)\Delta A+ f(4,10)\Delta A+ f(6,8)\Delta A + f(6,10) \Delta A + f(8,8) \Delta A}[/tex] Â Â
            [tex]\mathbf{+ f(8,10) \Delta A\Big)}[/tex]
[tex]\mathbf{Volume (V) =\Delta A\Big( f(4,8)+ f(4,10)+ f(6,8) + f(6,10)+ f(8,8) + f(8,10) }\Big)[/tex]
[tex]\mathbf{Volume (V) =4\Big( 32+40+48+60+64+80}\Big)[/tex]
Volume (V) = 4(324)
Volume (V) = 1296
Using the midpoint rule to estimate the volume of the solid;
The volume of the solid can be estimated by using the formula:
[tex]\mathbf{Volume (V) = \sum \limits^{m}_{i=1} \ \sum \limits^{n}_{j=1} \ f(x_1,y_j) \Delta A}[/tex]
where;
- the midpoint coordinates from the graph are: (3,7) , (3,9), (5,7) , (5,9) , (7,7) , (7,9)
[tex]\mathbf{Volume (V) =\Big( f(3,7)\Delta A+ f(3,9)\Delta A+ f(5,7)\Delta A + f(5,9) \Delta A + f(7,7) \Delta A+ f(7,9) \Delta A \Big)}[/tex]
[tex]\mathbf{Volume (V) =\Big( 4(21+27+35+45+49+63) \Big)}[/tex]
[tex]\mathbf{Volume (V) =\Big( 4(240)\Big)}[/tex]
Volume (V) Â = 960
Therefore, we can conclude that by using Riemann sum, the volume of the solid is 1296, and using the Midpoint rule, the volume of the solid is 960
Learn more about Riemann sum here:
https://brainly.com/question/23960718?referrer=searchResults
