Question

(a) Use a Reimann sum with\(m = n = 2,\)to estimate the value of , Where,\(R = [0,2] \times [0,1]\). Take the sample points to be upper right corners. (b) Use the Midpoint rule to estimate the integral in part (a).

Short answer

(a) Finding the estimated value of , where \(R = [0,2] \times [0,1]\) by using Riemann sum with \(m = n = 2\).

(b) Finding the estimated value of the integral in part (a) by using Midpoint Rule.

Step-by-step solution

Consider the integral:

\iint\limits_R {x{e^{ - xy}}dA} ; \(m = n = 2\)

Let \(R\) be the region \([0,2] \times [0,1]\). The region is to be divided into two subintervals in each direction. The sample point within each rectangle is to be in the upper right corner, as in the diagram below.

Finding the estimated integral value:-

From the diagram, the four sampling points are,\(\left( {1,\frac{1}{2}} \right),\left( {2,\frac{1}{2}} \right),(1,1){\rm{ }}\)and\((2,1)\).

Now,

Since\([a,b] = [0,2]\)and\(m = 2\)

Also,

Since\([c,d] = [0,1]\)and\(n = 2\)

The area of each rectangle is

With the region of integration partitioned and the sample points selected it is possible to estimate integral with a finite sum.

Therefore,

Finding   :-

(b)This is the same problem as in part (a) except that the sample points are to be the mid-points.

Since,

See the graph below.

The sample points are now\((0.5,0.25),(0.5,0.75),(1.5,0.25)\)and\((1.5,0.75)\).

Write the region of integration partitioned and the sample points selected it is possible to estimate the integral with a finite sum.

Therefore,