   Chapter 4.3, Problem 5E

Chapter
Section
Textbook Problem

# Sketch the area represented by g ( x ) . Then find g ′ ( x ) in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. g ( x ) = ∫ 1 x t 2 d t

To determine

To sketch and find:

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

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=ddx1xt2dt

Here ft=t2

Using part a) of Fundamental Theorem of Calculus we have

g'(x)=x2

Therefore, g'(x

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