Question

Estimate the volume of the solid that lies below the surface z=xy and above the rectangle R. Use a Riemann sum with m=3, n=2, and take the sample point to be the upper right corner of each square.
R = {(x,y): 0<=x<=6, 0<=y<=4}

Answer 1

(a) Using upper right endpoints for each of the little 'grids' yields (with f(x,y) = xy) 
∫∫R xy dA 
≈ 2 * 2 [f(2, 10) + f(4, 10) + f(6, 10) + f(2, 12) + f(4, 12) + f(6, 12)] 
= 4 [20 + 40 + 60 + 24 + 48 + 72] 
= 1056. 

(b) Using the midpoint for each of the little 'grids' yields 
∫∫R xy dA 
≈ 2 * 2 [f(1, 9) + f(3, 9) + f(5, 9) + f(1, 11) + f(3, 11) + f(5, 11)] 
= 4 [9 + 27 + 45 + 11 + 33 + 55] 
= 720. 

Answer 2

The volume of the solid that lies below the surface z=xy and above the rectangle R by using Riemann sum is 288

Given that:

• R = {(x, y)} : 0 ≤ x ≤ 6   , 0 ≤ y ≤ 4}.

                            ᵃ₁          ᵇ₁     ᵃ₂        ᵇ₂

Let consider f(x,y) = z; for which the surface z = xy

where;

• m = 3 and n = 2

Then, the width of the region can be estimated by using the formula:

• $\mathbf{\Delta x = \dfrac{b_1 -a_1}{m}}$

• $\mathbf{\Delta x = \dfrac{6 -0}{3}}$

• $\mathbf{\Delta x = 2}$

• $\mathbf{\Delta y = \dfrac{b_2 -a_2}{m}}$

• $\mathbf{\Delta y = \dfrac{4 -0}{2}}$

• $\mathbf{\Delta y = 2}$

The endpoints are;    

$\mathbf{x_o = a_1 = 0; }\\ \\ \mathbf{ x_1 =x_o+\Delta x=2} \\ \\ \mathbf{x_2 =x_1+\Delta x=4 } \\ \\ \mathbf{x_3 =x_2+\Delta x = 6}$

$\mathbf{y_o = a_2 = 0; }\\ \\ \mathbf{ y_1 =y_o+\Delta y=2} \\ \\ \mathbf{y_2 =y_1+\Delta y=4 } }$

The upper right regions are; { (2,2) , (2,4) ,  (4,2) ,  (4,4) , (6,2) , (6,4) }

Using the sample point to be the upper right corner of each square, the upper right endpoint is:

$\mathbf{\iint_{r \ xydA} =\Delta x \times \Delta y} \mathbf{\Big[f(2,2)+f(2,4)+f(4,2)+f(4,4)+f(6,2)+f(6,4)\Big]}}$

$\mathbf{=2\times 2} \mathbf{\Big[(2\times 2)+(2\times4)+ (4\times2)+ (4 \times 4)+(6 \times 2)+(6 \times4 )\Big]}}$

$\mathbf{=4} \mathbf{\Big[4+8+8+ 16+12+24\Big]}}$

$\mathbf{=4} \mathbf{\Big[72\Big]}}$

= 288

Learn more about Riemann sum here:

https://brainly.com/question/23960718?referrer=searchResults