   Chapter 4, Problem 32RE

Chapter
Section
Textbook Problem

Finding Area by the Limit Definition In Exercises 29-32, use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region. y = 1 4 x 3 ,         [ 2 , 4 ] .

To determine

To calculate: The x-axis over the given interval [ 2,4 ] using the limit process for y=14x3.

Explanation

Given:

y=14x3 in the interval [ 2,4 ]

Formula used:

S(n)=i=1nf(Mi)Δx, Δx=ban and A=limnS(n)

Calculation:

Consider the given function y=14x3 in the interval [ 2,4 ].

Divide the interval [ 2,4 ] in to n subinterval, each of which has fixed width, Δx=42n=2n

Therefore, the right end points are, Mi=2+i(2n)=2in+2

Calculate the upper sum of the region using the right end points, S(n)=i=1nf(Mi)Δx=i=1nf(2in+2)(2n)=2ni=1n{ 14(2in+2)3 }=4ni=1n{ (in+1)3 }

This is further simplified as, S(n)=4ni=1n(i3n3+3i2n2+

To determine

To graph: The area of the region bounded by the graph of the function y=14x3

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