The area of the region is an illustration of definite integrals.
The approximation of the area of the region R is 9.98
The given parameters are:
[tex]\mathbf{f(x) = \frac 7x}[/tex]
[tex]\mathbf{Interval = [1,3]}[/tex]
[tex]\mathbf{n = 4}[/tex] ------ sub intervals
Using 4 sub intervals, we have the partitions to be:
[tex]\mathbf{Partitions = [1,\frac 32]\ u\ [\frac 32, 2]\ u\ [2,\frac 52]\ u\ [\frac 52,3]}[/tex]
List out the right endpoints
[tex]\mathbf{x= \frac 32, 2 ,\frac 52, 3}[/tex]
Calculate f(x) at these partitions
[tex]\mathbf{f(3/2) = \frac{7}{3/2} = \frac{14}{3}}[/tex]
[tex]\mathbf{f(2) = \frac{7}{2}}[/tex]
[tex]\mathbf{f(5/2) = \frac{7}{5/2} = \frac{14}{5}}[/tex]
[tex]\mathbf{f(3) = \frac{7}{3}}[/tex]
So, the approximated value of the definite integral is:
[tex]\mathbf{\int\limits^3_1 {f(x)} \, dx \approx \frac{3}{4}(\sum f(x))}[/tex]
This becomes
[tex]\mathbf{\int\limits^3_1 {f(x)} \, dx \approx \frac{3}{4}(\frac{14}{3} + \frac{7}{2} + \frac{14}{5} + \frac{7}{3})}[/tex]
[tex]\mathbf{\int\limits^3_1 {f(x)} \, dx \approx \frac{3}{4}(4.67 + 3.50 + 2.80 + 2.33)}[/tex]
[tex]\mathbf{\int\limits^3_1 {f(x)} \, dx \approx \frac{3}{4}(13.30)}[/tex]
Expand
[tex]\mathbf{\int\limits^3_1 {f(x)} \, dx \approx 9.98}[/tex]
Hence, the approximation of the area of the region R is 9.98
Read more about definite integrals at:
https://brainly.com/question/9897385
