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As we will see in Section 6.3, the definite integral

L=ab1+f'x2dx

calculates the length of the curve of fxfrom x=ato x=b. Explain intuitively what this might have to do with the distance formula calculation.

Short Answer

Expert verified

Therefore, the definite integral L=ab1+f'x2dxmight have done with the distance formula calculation is explained.

Step by step solution

01

Step 1. Given information

L=ab1+f'x2dx.

02

Step 2.We can use the distance formula to calculate the length lk of each line segment.

Let us consider a graph.

wherex=b-an,xk=a+kx,yk=fxk-fxk-1for allk=0,1,2,3.....,n.

03

Step 3. lk= Length of the kth line segment.

lk=x2+yk2=x21+yk2x2=x1+ykx2

04

Step 4.Now given a function fx, an interval a,b, 

And a number of subdivisions n, we have defined n subintervals, each with a line segment that approximates a small piece of the curve. Adding up the lengths of these line segments gives us an approximation for the length of the curve:

Length of fxon role="math" localid="1649144454037" a,bk=1nlk

=k=1n1+ykx2.x.

[As n,ykx=f'x]

role="math" localid="1649144736719" =limnk=1n1+f'x2x=ab1+f'x2dx

Thus, the definite integral L=ab1+f'xdxmight have to done with the distance formula calculation.

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