#### To determine

A region whose area is given

#### Answer

*fx=x, 1≤x≤4*

#### Explanation

**1) Concept:**

The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles.

*A=limn→∞Rn=limn→∞fx1∆x+fx2∆x+…+fxn∆x=limn→∞∑i=1nfxi∆x*

The width of the interval *a, b* is *b-a*, so the width of each *n* strip is

*∆x=b-an*

where *x0=a * and * xn=b *. The right end points of the subintervals are *xi=a+i∆x*.

**2) Given:**

*limn→∞∑i=1n3n1+3in*

3) **Calculation:**

It is given that

*A= limn→∞∑i=1n3n1+3in*

The general formula of area is

*A=limn→∞∑i=1nfxi∆x*

Comparing these equations to each other,

*fxi=1+3in*

and

*∆x=3n*

Now,

*a+i∆x=1+3in*

Therefore, *a=1*, so *b=4* since

* ∆x=b-an*

and

*xi=1+i·3n*

This means *fxi=xi*

That means the required region is

*fx=x, 1≤x≤4*

**Conclusion:**

The region is *fx=x, 1≤x≤4*