#### To determine

**To estimate:**

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

#### Answer

A=2434

#### 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=x3, 0≤x≤27

**3) Calculation:**

The graph of the given curve y=x3, 0≤x≤27 bounded between x=0 and x=27 is

The region enclosed by the curves y=x3, x=0, x=27 and y=0 is shaded in black

A rough estimate of area can be obtained by counting the number of small squares. Notice that 25 squares make my 5 unit of area. That is 5 squares make 1 unit of area. There are about 311 squares. So an estimate of area is 321/5 sq unit. To find exact area

From the above graph, the curve y=fx=x3 bounded between x=0 and x=27 is continuous on 0,27, then

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

∫027x3 dx=F27-F0…(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),

ddxx13+113+1=x3=x13

ddxx4343=x3

From (2),

Fx=x4343=34x43

Substitute F(x) in (1) at x=0 and x=27,

∫027x3 dx=F27-F(0)

=342743-34043

=343343-0

=3434

=354

=2434

**Conclusion:**

Therefore, the required area A=2434