   Chapter 10.2, Problem 50E

Chapter
Section
Textbook Problem

In Exercise 10.1.43 you were asked to derive the parametric equations x = 2a cot θ, y = 2a sin2θ for the curve called the witch of Maria Agnesi. Use Simpson’s Rule with n = 4 to estimate the length of the arc of this curve given by π/4 ≤ θ ≤ π/2.

To determine

To estimate: The length of the curve using Simpson’s Rule for the parametric equations of x=2acotθ and y=2asin2θ.

Explanation

Given:

The parametric equation for the variable x is as below.

x=2acotθ

The parametric equation for the variable y is as below.

y=2asin2θ

The range of θ varies from π4 to π2.

Calculation:

Differentiate the parametric equation x with respect to t.

x=2acotθdxdθ=2acsc2θ

Differentiate the parametric equation y with respect to t.

y=2asin2θdydθ=4asinθcosθdydθ=2asin2θ

The length of the curve is, L=π4π2(dxdt)2+(dydt)2dt

Substitute (2asin2θ) for dydt and (2acsc2θ) for dxdt in the above equation.

L=π4π2(2acsc2θ)2+(2asin2θ)2dθ=π4π24a2csc4θ+4a2sin22θdθ=2aπ4π2csc4θ+sin22θd

Compute the value of Δt using the formula.

Δt=ban

Substitute the value of (π2) for b, (π4) for a, and 4 for n in equation.

Δθ=π2+π44=π16

The subintervals are (π4,5π16),(5π16,6π16),(6π16,7π16),(7π16,π2)

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