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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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Most popular questions from this chapter

Problem Zero: Read the section and make your own summary of the material.

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

x3x2+1dx

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingx2+45/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2a2.

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

Domains and ranges of inverse trigonometric functions: For each function that follows, (a) list the domain and range, (b) sketch a labeled graph, and (c) discuss the domains and ranges in the context of the unit circle.

f(x)=sec1x

Suppose v(x) is a function of x. Explain why the integral

of dv is equal to v (up to a constant).

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