Question

(a) Write an integral which represents the circumference of a circle of radius $r$ (b) Evaluate the integral, and show that you get the answer you expect.

Name: Problem 52, Chapter 8: Calculus
Uploaded: 2024-01-23T00:28:51-08:00
Channel: Victor Salazar
Description: (a) Write an integral which represents the circumference of a circle of radius r (b) Evaluate the integral, and show that you get the answer you expect.

   (a) Write an integral which represents the circumference of a circle of radius $r$
(b) Evaluate the integral, and show that you get the answer you expect.

Calculus

Deborah… 5th Edition

Chapter 8, Problem 52

Instant Answer

Solved by Expert Victor Salazar

01/23/2024

Step 1

To represent this as an integral, we can think of the circumference as the sum of infinitely small line segments along the circle. Each line segment has a length of $ds$, which is an infinitesimally small arc length. We can integrate these line segments over the Show more…

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(a) Write an integral which represents the circumference of a circle of radius $r$ (b) Evaluate the integral, and show that you get the answer you expect.

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Key Concepts

Arc Length of a Curve

The arc length of a curve is determined by integrating the square root of the sum of the squares of the derivatives of the curve’s coordinates with respect to a parameter. This formula allows us to calculate the length of curves that are not straight lines, making it a fundamental concept in calculus.

Parametrization of a Curve

Expressing a curve via a parameter, such as using trigonometric functions for circles, simplifies many calculations. With a proper parameterization, the derivatives needed for the arc length formula become straightforward, enabling the setup of integrals that compute geometric quantities.

Definite Integration

Definite integration is used to accumulate small quantities over a specific interval, allowing the exact calculation of values such as lengths, areas, and volumes. By applying the fundamental theorem of calculus, one can evaluate an integral representing a continuous sum along a curve to obtain a precise result.

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