Find the length of the curve y=(4-x^2 / 3)^3 / 2 over the interval [0,8] .

Name: Problem 45, Chapter 7: Calculus Early Transcendentals
Uploaded: 2020-08-31T19:10:40-08:00
Duration: 3 min 12 s
Channel: Nick Johnson
Description: Find the length of the curve y=(4-x^2 / 3)^3 / 2 over the interval [0,8] .

step 1 Okay, so we know that the length of a curve, F of X, over an interval A, B is, well, is given by

Find the length of the curve y=(4-x^2 / 3)^3 / 2 over the...

Key Concepts

Integration Techniques

Integration techniques, including substitution, rationalizing expressions, or other methods, are crucial for solving integrals that arise from applying the arc length formula. These methods simplify complex integrals into forms that can be more readily evaluated. Mastery of integration techniques is important for handling and solving integrals involving composite functions, radicals, and fractional exponents that naturally occur in problems involving curve lengths.

Arc Length Formula

The arc length formula is a fundamental concept in calculus that allows one to determine the length of a curve defined by a function. It involves integrating the square root of (1 plus the square of the derivative of the function) over a given interval. This formula is key in converting a geometric problem into an integral calculus problem, making it possible to compute lengths of curves even when they are not straight.

Chain Rule in Differentiation

The chain rule is an essential technique in calculus used to differentiate composite functions. When a function is composed of one function inside another, the chain rule provides a method to compute the derivative by differentiating the outer function and then multiplying it by the derivative of the inner function. This technique is particularly useful when the function involves nested expressions, such as functions with exponents or radicals.

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