Question

If\(R = (0,4) \times ( - 1,2)\), use a Reimann sum with\(m = 2,n = 3\)to estimate the value of Take the sample points to be (a) The lower right corners and (b) the upper left corners of the rectangles.

Short answer

Finding thelower rightcorners and theupper leftcorners of therectanglesbytaking sample points.

Step-by-step solution

Step 1:

Consider the following integral:

.

Here,\(R = (0,4) \times ( - 1,2)\)

Estimate the value of the integral by using Reimann sum with

1. The objective is to estimate the value of the above integral take the sample points as the lower right corners. Approximate the integrals by using a double Riemann sum is as follows:

Here\(m\)represents the number of partitions along the\(x\)interval\((a,b)\).

\(a = {x_0} < {x_1} < ... < {x_{i - 1}} < {x_i} < ... < {x_m} = b\)

Here,\(n\)represents the number of partitions along the\(y\)interval\((c,d)\).

\(c = {y_0} < {y_1} < ... < {y_{i - 1}} < {y_i} < ... < {y_n} = d\)

And

Consider the interval for  \(x\) is \((0,4)\):-

For

Similarly, consider the interval for\(y\)is\(( - 1,2)\)

For

Then,

the integral as the double Riemann sum,

Consider the following figure:-

From the above figure, the lower right corner points are as follows:

Finding the estimated value of the integral :-

Consider the function,\(f(x,y) = 1 - x{y^2}\)

The function value of all these points is,

\(\begin{aligned}{c}f(2, - 1) = 1 - 2{( - 1)^2}\\ = 1 - 2\end{aligned}\)

\(\begin{aligned}{c}f(2,0) = 1 - 2{(0)^2}\\ = 1 - 0\end{aligned}\)

\(\begin{aligned}{c}f(2,1) = 1 - 2{(1)^2}\\ = 1 - 2\end{aligned}\)

\(\begin{aligned}{c}f(4, - 1) = 1 - 4{( - 1)^2}\\ = 1 - 4\end{aligned}\)

\(\begin{aligned}{c}f(4,0) = 1 - 4{(0)^2}\\ = 1 - 0\end{aligned}\)

\(\begin{aligned}{c}f(4,1) = 1 - 4{(1)^2}\\ = 1 - 4\end{aligned}\)

Use equation\((2)\)estimate the integral as follows:

\(\)

Therefore, the estimated value of the integral is .

The upper left corner points are as follows:-

\((0,0),(2,0),(0,1),(2,1),(0,2),(2,2)\)

Consider the function,

The value of the function at these points is,

\(\begin{aligned}{c}f(0,0) = 1 - 0{(0)^2}\\ = 1 - 0\end{aligned}\)

\(\begin{aligned}{c}f(0,1) = 1 - 0{(1)^2}\\ = 1 - 0\end{aligned}\)

\(\begin{aligned}{c}f(0,2) = 1 - 0{(2)^2}\\ = 1 - 0\end{aligned}\)

\(\begin{aligned}{c}f(2,0) = 1 - 2{(0)^2}\\ = 1 - 0\end{aligned}\)

\(\begin{aligned}{c}f(2,1) = 1 - 2{(1)^2}\\ = 1 - 2\end{aligned}\)

\(\begin{aligned}{c}f(2,2) = 1 - 2{(2)^2}\\ = 1 - 8\end{aligned}\)

Finding estimated value of the integral:

Use equation\((2)\)estimate the integral as follows:

Therefore, the estimated value of the integral is .