Chapter 8, Problem 9RE

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Find the length of the curve y = ∫ 1 x t − 1   d t   1 ≤ x ≤ 16

To determine

To find: The value of the length of the curve.

Explanation

Given information:

The function of the curve is y=âˆ«1x(tâˆ’1)dtâ€‰â€‰1â‰¤xâ‰¤16 (1)

The lower limit is 1 and the upper limit is 16.

The Fundamental Theorem of Calculus (Part 1) is shown below:

g(x)=âˆ«axf(t)dt aâ‰¤xâ‰¤b

Here, the continuous function on the interval [a,â€‰b] is f and the function is g.

Condition for the theorem to be valid:

• If the function is continuous on the interval [a,â€‰b] and differentiable on (a,â€‰b) .
• Also for the continuous condition, g'(x)=f(x) .

Calculation:

Apply Fundamental Theorem of Calculus (Part 1) in Equation (1).

y'(x)=xâˆ’1dydx=xâˆ’1

The expression to find the length of the curve (L) is shown below:

L=âˆ«ab1+(dydx)2â€‰dx (2)

Here, the derivative of the function y is dydx , the lower limit is a, and the upper limit is b

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th