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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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 + 2 , 0 , 1

To determine

To calculate: Area of the region bounded by y=x2+2, in the interval [0,1] and the x-axis.

Explanation

Given: y=x2+2, in the interval [0,1].

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

Use 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 [0,1].

Partition the interval into n subintervals each of width, Δx:

Δx=10n=1n

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

Right endpoints (Mi) are:

0+in=in,

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

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

Use value of y(in),

Area=limni=1n((in)2+2)(1n)=limni=1n1n3(i2)+limni=1n(2n)=lim</

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