   Chapter 4, Problem 3PS

Chapter
Section
Textbook Problem

Evaluating a Sum and a Limit In Exercises 3 and 4, (a) write the area under the graph of the given function defined on the given interval as a limit. Then (b) evaluate the sum in part (a), and (c) evaluate the limit using the result of part (b). y = x 4 − 4 x 3 + 4 x 2 ,         [ 0 ,   2 ] ( H i n t : ∑ i = 1 n i 4 = n ( n + 1 ) ( 2 n + 1 ) ( 3 n 2 + 3 n − 1 ) 30 )

(a)

To determine

To calculate: An expression for the area under the graph of the function y=x44x3+4x2 over the interval [0,2].

Explanation

Formula Used:

The formula for the approximate area under the curve y=f(x) over the interval [a,b] is:

A=limni=1nf(ci)Δx

Here, Δx=ban is the step size and ci=(ba)in.

Calculation:

For the curve y=x44x3+4x2, over the interval [0,2],

Δx=2nci=2in

The expression for the area under the curve would be:

A=limni=1nf(2in)(2n)=limni=1n[((

(b)

To determine

To calculate: The simplified expression for the sum of area under the curve.

(c)

To determine

To calculate: The area under the curve.

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