Question

(a) Estimate the volume of the solid that lies below the surface\(z = xy\)and above the rectangle\(R = \{ (x,y)|0 \le x \le 6,0 \le y \le 4\} \). Use a Reimann sum with\(m = 3,n = 2,\)and a regular partition, 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).

Short answer

(a) Estimating the volume of the solid that lies below the surface \(z = xy\) and above the rectangle \(R = \{ (x,y)|0 \le x \le 6,0 \le y \le 4\} \). Using Reimann Sum with \(m = 3,n = 2\) and a regular partition.

(b) Estimating the volume ofthe solid in part(a)by using the Midpoint Rule.

Step-by-step solution

Plotting the graph :-

Consider the solid that lies below the surface\(z = xy\)and above the rectangle\(R\)defined as below.

\(R = \{ (x,y)|0 \le x \le 6,0 \le y \le 4\} \).

Figure below shows the solid as defined above.

Diagram !!!!

Finding volume of the solid using Reimann sum with \(m = 3,n = 2\):-

1. The objective is to estimate the volume of the solid by using a Riemann sum with\(m = 3,n = 2.\)

For Riemann sum, the squares with\(m = 3,n = 2\)are shown in the below figure.

Diagram!!!!

Area of each square with side length\(2\)is\({2^2} = 4\)

This implies\(\Delta A = 4.\)

The upper right corner of each square is\((2,2),(2,4),(4,2),(4,4),(6,2),(6,4).\)

And the function is

The volume of the solid using Riemann sum with\(m = 3,n = 2\)is

$\begin{gathered}

V = \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {({x_1},{y_1})\Delta A} } \hfill \\

\Rightarrow V = f(2,2)\Delta A + f(2,4)\Delta A + f(4,2)\Delta A + f(4,4)\Delta A + f(6,2)\Delta A + f(6,4)\Delta A \hfill \\

\Rightarrow V = (2)(2)(4) + (2)(4)(4) + (4)(2)(4) + (4)(4)(4) + (6)(2)(4) + (6)(4)(4) \hfill \\

\Rightarrow V = 16 + 32 + 32 + 64 + 48 + 96 \hfill \\

\Rightarrow \boxed{V = 288} \hfill \\

\end{gathered} $

Therefore, the required volume of the solid using Riemann sum is square units.

Finding points :-

1. The objective is to estimate the volume of the solid using Midpoint rule with\(m = 3,n = 2\). Midpoints of the six squares are shown in the below figure.

Diagram !!!!

Area of the square with side length\(2\)is\({2^2} = 4.\)This implies\(\Delta A = 4\).

The Midpoint of each square is \((1,1),(3,1),(5,1),(1,3),(3,3),(5,3).\) And the function is

Finding Volume of the solid using Midpoint rule with \(m = 3,n = 2\)is as follows:

Therefore, the required volume of the solid using the Midpoint rule is square units.

Hence the required volume of solid using Reimann sum is \(288\) square units and the required volume of the solid using the midpoint rule is \(144\) square units.