#### To determine

**To sketch and find:**

The area represented by g(x) and find the derivative by using Fundamental Theorem of Calculus.

#### Answer

Graph is,

By using Fundamental Theorem of Calculus a) part, g'(x)=x2

By using Fundamental Theorem of Calculus b) part, g'(x)=x2

#### Explanation

**1) Concept:**

The Fundamental Theorem of Calculus-Suppose f is continuous on [a, b] then

a) If gx=∫axftdt, then g'=fx

b) ∫abfxdx=Fb-F(a), where F is an Antiderivative of f i.e F'=f

**2) Given:**

gx=∫1xt2dt

3) **Calculation:**

The function defined gx=∫1xt2dt is the area between graph of fx=x2 and x axis from x = 1 to x = x

Now,

gx=∫1xt2dt

Take the derivative of both sides

g'x=ddx∫1xt2dt

Here ft=t2

Using part a) of Fundamental Theorem of Calculus we have

g'(x)=x2

Therefore, g'(x)=x2

Now, use Fundamental Theorem of Calculus part b)

We have,

gx=∫1xt2dt

Let ft=t2

The antiderivative of f is G(t)=13t3

Use Fundamental Theorem of Calculus part b) to get

gx=∫1xt2dt=Gx-G(1)

Use G(t)=13t3 in the above equation to get

gx=13x3-13(1)3

Simplify

gx=13x3-13

Taking derivative

g'(x)=13(3x3-1)-0

Simplifying

g'(x)=x2

Therefore, g'(x)=x2

**Conclusion:**

Therefore, by using Fundamental Theorem of Calculus part a) g'(x)=x2

and by using Fundamental Theorem of Calculus part b), g'(x)=x2