#### To determine

**To estimate:**

The rough area of the region forthe given curve then find exact area

#### Answer

A=215648

#### Explanation

**1) Concept:**

i) Fundamental theorem of Calculus, Part 2

If f is continuous on a,b, then

∫abfxdx=Fb-F(a)

where F is any antiderivative of f, that is, a function F such that F'=f

ii) Power rule for antiderivative:

ddxxn+1n+1=xn

**2) Given:**

y=x-4, 1≤x≤6

**3) Calculation:**

The curves y=x-4, 1≤x≤6 are given by

From above graph, the given curve y=x-4, 1≤x≤6 bounded between x=1 and x=6 is

The region enclosed by the curve y=x-4, x=1, x=6 and y=0 is

Here 50 small squares make up one unit square area. There are about 17 small squares (partial squares included) within the shaded region. So a rough estimate of area is 17/50 = 0.34 sq unit. To find the exact area of the region enclosed by the given curves;From above graph, the curve y=fx=x-4 bounded between x=1 and x=6, is continuous on 0,6 then

By using concept i) (Fundamental theorem of Calculus, Part 2),

∫16x-4 dx=F6-F1…(1)

Where F is antiderivative of f, that is, a function F such that F'(x)=f(x) means

ddxFx=fx…(2)

By using concept ii) (power rule of antiderivative),

ddxx-4+1-4+1=x-4

ddxx-3-3=x-4

From (2),

Fx=x-3-3=-13x3

Substitute F(x) in (1) at x=1 and x=6,

∫16x-4 dx=F6-F(1)

=-1363--1313

=-13216--13

=-1648+13

=-1216·3+13

Factoring out 13 as the common factor

=13-1216+1

=13-1+216216

=13215216

=215648

**Conclusion:**

Therefore, the required area is A=215648