Find the exact length of the curve. y^2=4(x+4)^3, 0 ⩽x ⩽2, y>0

Name: Problem 8, Chapter 8: Calculus
Uploaded: 2020-05-30T09:12:22-08:00
Duration: 2 min 58 s
Channel: Rakesh Kumar Sharma
Description: Find the exact length of the curve. y^2=4(x+4)^3, 0 ⩽x ⩽2, y>0

step 1 Our integration, lower limit to upper limit, so lower limit is 0, upper limit is 2. So it will b

Find the exact length of the curve. y^2=4(x+4)^3, 0 ⩽x ⩽2,...

Key Concepts

Arc Length Formula

The arc length formula is essential in determining the length of a curve over a given interval. For a curve expressed as y = f(x), the formula L = ?[a,b] sqrt(1 + (dy/dx)²) dx is used, where dy/dx is the derivative of the function. This concept is central to problems involving the distance along a curve and underpins the method of setting up and evaluating the necessary integral.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In the context of arc length problems, computing the derivative of the function (dy/dx) accurately is crucial, as the derivative appears squared inside the integral for the curve’s length. Techniques like the chain rule and power rule are typically applied here.

Integration Techniques

Integration techniques are used to evaluate the integral that arises from the arc length formula. In many cases, the integral involves complex expressions such as square roots of algebraic functions, which may require methods like substitution or algebraic manipulation to simplify before integration. Mastery of these techniques is vital for finding exact expressions in calculus problems.

Substitution

Substitution is an integration method that simplifies a difficult integral by transforming it into an easier form. This is achieved by changing variables, which often simplifies the integrand into a standard form that is more straightforward to integrate. This technique is particularly useful when the integrand is a composite function that results from differentiating a power or applying the chain rule.

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