   Chapter 4.2, Problem 52E

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 = 4 − x 2 , [ − 2 , 2 ]

To determine

To calculate: Area of the region bounded by y=4x2, in the interval [2,2] and the x-axis.

Explanation

Given: y=4x2, in the interval [2,2]

Formula used: Formula for the sum of squares of first n natural:i=1ni2=n(n+1)(2n+1)6

The sum of a constant n times is written as:i=1nc=nc

Formula for the sum of first n natural numbers:i=1ni=n(n+1)2

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 [2,2].

Partition the interval into n subintervals.

Δx=2(2)n=4n.

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

Right endpoints (Mi) are:2+4in=4i2nn

So, the area will be:

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

where, Mi=4i2nn.

Therefore, Area=limni=1ny(4i2nn)(Δx)

Use value of y(4i2nn) and Δx,

Area=limni=1n(4(4i2nn)2)(4n)

Split the expression in parts to use summation formulas:

Area=limni=1n16nlimni=1n4n3(4i2n)2=limni=1n16nlimni=1n4n3

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