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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.2.25
Chapter 11, Problem 11.2.25

Lengths of Curves


Find the lengths of the curves in Exercises 25–30.


x = cos t, y = t + sin t, 0 ≤ t ≤ π

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Recall the formula for the length of a curve defined parametrically by \(x = x(t)\) and \(y = y(t)\) over the interval \(a \leq t \leq b\): \[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]
Identify the given parametric functions: \[x(t) = \cos t, \quad y(t) = t + \sin t\] and the interval: \[0 \leq t \leq \pi\]
Compute the derivatives of \(x(t)\) and \(y(t)\) with respect to \(t\): \[\frac{dx}{dt} = -\sin t\] \[\frac{dy}{dt} = 1 + \cos t\]
Substitute the derivatives into the arc length formula under the square root: \[\sqrt{(-\sin t)^2 + (1 + \cos t)^2} = \sqrt{\sin^2 t + (1 + \cos t)^2}\]
Simplify the expression inside the square root as much as possible to prepare for integration, then set up the integral for the length: \[L = \int_0^{\pi} \sqrt{\sin^2 t + (1 + \cos t)^2} \, dt\] From here, you can proceed to simplify and evaluate the integral to find the length.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. In this problem, x and y are given in terms of t, allowing the curve to be analyzed by studying these functions over the specified interval.
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Parameterizing Equations

Arc Length Formula for Parametric Curves

The length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is found using the integral of the square root of the sum of the squares of the derivatives: L = ∫_a^b √[(dx/dt)² + (dy/dt)²] dt. This formula calculates the total distance traveled along the curve.
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Arc Length of Parametric Curves

Differentiation of Trigonometric Functions

To apply the arc length formula, derivatives of x(t) and y(t) must be computed. Understanding how to differentiate trigonometric functions like sine and cosine is essential, as these derivatives determine the integrand for the arc length calculation.
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Related Practice
Textbook Question

Eccentricities and Directrices


Exercises 29–36 give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section.


e = 2, x = 4

Textbook Question

Tangent Lines to Parametrized Curves


In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.


x = t + eᵗ, y = 1 − eᵗ, t = 0

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Textbook Question

Examples of Polar Equations


[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.


r = −2 cos θ

Textbook Question

Cartesian to Polar Equations


Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.


x - y = 3

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Textbook Question

Polar to Cartesian Equations


Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.


r² = 4r sin θ

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Textbook Question

Graphing Sets of Polar Coordinate Points


Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises 11–26.


θ = π/2, r ≥ 0

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