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CalculusCalculus10th Edition

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Chapter 8, Problem 72RE
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Arc Length In Exercises 71 and 72, approximate to two decimal places the are length of the curve over the given interval.

F u n c t i o n                                                 I n t e r v a l                                     7 2 .   y = sin n 2 x                 [ 0 , π ]

To determine

To calculate: The arc length of the curve y=sin2x on the interval [0,π] upto two decimal places.

Explanation

Given:

The curve y=sin2x on the interval [0,π].

Formula used:

Trapezoidal Rule,

∫abf(x)dx≈Δx2(f(x0)+2f(x1)+2f(x2)+...+2f(xn−1)+f(xn))

Where Δx=b−an,xi=a+iΔx,i=0,1,2....

Calculation:

Consider the curve y=sin2x on the interval [0,π].

Now, the arc length of a curve is given by, A=∫0π(1+(dydx)2)dx.

Differentiate y=sin2x with respect to x as follows,

y=sin2xdydx=2sinxcosx=sin2x

Then,

A=∫0π1+sin22xdx

Recall Trapezoidal rule.

∫abf(x)dx≈Δx2(f(x0)+2f(x1)+2f(x2)+...+2f(xn−1)+f(xn))

Compare ∫0π1+sin22xdx to ∫abf(x)dx

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P-15PSCh-8 P-16PSCh-8 P-17PSCh-8 P-18PSCh-8 P-19PSCh-8 P-20PSCh-8 P-21PSCh-8 P-22PS

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