Find the exact length of the curve. y=(x^3)/(3)+(1)/(4 x), 1 ⩽x ⩽2

Name: Problem 9, Chapter 7: Essential Calculus Early Transcendentals
Uploaded: 2020-05-29T18:55:53-08:00
Duration: 3 min 27 s
Channel: Rakesh Kumar Sharma
Description: Find the exact length of the curve. y=(x^3)/(3)+(1)/(4 x), 1 ⩽x ⩽2

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Find the exact length of the curve. y=(x^3)/(3)+(1)/(4 x), ...

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Key Concepts

Arc Length Formula

This formula provides a method to determine the length of a curve defined by a function over a specific interval. It involves integrating the square root of one plus the square of the derivative of the function, typically expressed as L = ?[a to b] ?(1 + (dy/dx)²) dx. Understanding this formula is essential for measuring distances along curved paths in calculus and geometric applications.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the instantaneous rate of change or slope of the curve at any point. In the context of arc length, obtaining the derivative is a critical step because it is used to compute (dy/dx)² inside the integral. Mastery of differentiation techniques, such as the power rule and rules for differentiating rational functions, is fundamental.

Integration Techniques

Integration techniques are vital for evaluating the definite integral that arises in the arc length formula. These techniques, which may include algebraic manipulation, substitution, and sometimes trigonometric substitution, help simplify the square root expression and enable the exact calculation of the length of the curve. They form an integral part of solving problems in integral calculus.

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