Functions and Arc Length Calculations

The video tutorial explains how to calculate the arc length of a curve using integration. It introduces the formula for arc length and demonstrates its application through multiple examples. The tutorial covers different scenarios, including functions in terms of X and Y, and provides detailed steps for solving each example. The video aims to enhance understanding of arc length calculation in calculus.

What is the formula used to calculate the arc length of a curve?

• Differentiation of the function

• Integration of the square root of 1 plus the derivative squared

• Integration of the derivative

• Multiplication of the function by its derivative

In the first example, what substitution is made to simplify the integration?

• U = 81x

• U = 1 + x

• U = 1 + 81x

• U = x^2

What are the new limits of integration after substitution in the first example?

• 0 to 1

• 1 to 81

• 1 to 82

• 0 to 82

In the second example, what is the function given for calculating the arc length?

• Y = 3x^2

• Y = 3x^(2/3)

• Y = x^3

• Y = 2x^(3/2)

What is the result of squaring the derivative in the second example?

• x^(1/3)

• x^(5/3)

• x^(2/3)

• x^(4/3)

In the third example, what is the function given in terms of y?

• X = (1/3)(y^3 + 2)^(2/3)

• X = (1/2)(y^3 + 2)^(3/2)

• X = (1/3)(y^2 + 2)^(3/2)

• X = (1/3)(y^2 + 3)^(2/3)

What is the perfect square trinomial identified in the third example?

• y^2 + 2y + 1

• y^2 + 3y + 1

• y^2 + y + 1

• y^2 + 2y + 2

In the final example, what is the first step in simplifying the function?

• Factoring out the GCF

• Finding the derivative

• Rewriting the square root

• Distributing y

What is the GCF factored out in the final example?

• 1/4 y^(1)

• 1/2 y^(1/2)

• 1/3 y^(1/2)

• 1/4 y^(1/2)

What is the final answer for the arc length in the last example?

• 32/3

• 76/3

• 11/3

• 64/3