   Chapter 4.2, Problem 55E

Chapter
Section
Textbook Problem

Finding Area by the Limit Definition In Exercises 47-56, 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 = x 2 − x 3 , [ − 1 , 1 ]

To determine

To calculate: Area of the region bounded by y=x2x3, in the interval [1,1] and the x-axis.

Explanation

Given: y=x2x3, in the interval [1,1]

Formula used: Formula for the sum of cubes of first n natural numbers:

i=1ni3=n2(n+1)24

Formula for the sum of first n natural numbers:

i=1ni=n(n+1)2

Formula for the sum of squares of first n natural numbers:

i=1ni2=n(n+1)(2n+1)6

Using right endpoints area is written as:

Area=limni=1ny(Mi)(Δx), where Mi are the right endpoints.

Calculation: Function y is continuous and non-negative in the interval [1,1].

Partition the interval into n subintervals each of width Δx:

Δx=1(1)n=2n

Area can be calculated by left endpoints (mi) or right endpoints (Mi).

Right endpoints (Mi) are:1+2in=2inn,

i=1,2,3,.......,n

So,

Area=limni=1ny(Mi)(Δx) where, Mi are the right endpoints, and Mi=2inn.

So,

Area=limni=1ny(2inn)(Δx)

Use value of y(2inn) and Δx,

Area=limni=1n((2inn)2(2inn)3)(2n)

Split the expression in parts to use summation formula:

Area=limni=1n2(2in)2n3limni=1n2(2in)3n4

Expand the first expression, and second expression:

Area=limni=1n2(4i2+n24in)n3limni=1n2(8i3n312i2n+6in2)n4

Factor out 2n3 from the first sum and 2n4

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