Calculus and Trigonometry Concepts

The video tutorial explains how to evaluate a line integral along a curve C, parameterized by R of T, over a closed interval from 0 to 2. It covers the process of expressing the function x^2 + y^2 + z^2 in terms of T, finding derivatives using the chain rule, and substituting these into the integral. The tutorial then simplifies the integral using trigonometric identities and performs the final integration to find the solution, which is presented as 364√3/3.

What is the line integral evaluated with respect to in the given problem?

• Arc Length s

• Time t

• Angle θ

• Distance d

Which of the following is the correct expression for X(T) in the parametric equations?

• 4T

• 3 cos 2T

• 2T

• 3 sin 2T

What is the derivative of Z(T) with respect to T?

• 2

• 4

• 6 sin 2T

• 3 cos 2T

After substituting X, Y, and Z, what is the expression for x^2 + y^2 + z^2?

• 9 cos^2 2T + 16T^2

• 9 sin^2 2T + 16T^2

• 9 cos^2 2T + 9 sin^2 2T + 16T^2

• 16T^2

What is the simplified expression for the square root of the sum of the squares of the derivatives?

• √52

• √36

• √16

• √9

What is the factorization of √52 used in the simplification?

• 2√13

• 4√13

• 3√13

• 5√13

Which trigonometric identity is used in the simplification process?

• 1 + cot^2 θ = csc^2 θ

• sin 2θ = 2 sin θ cos θ

• tan^2 θ + 1 = sec^2 θ

• cos^2 θ + sin^2 θ = 1

What is the integral of 9 + 16T^2 with respect to T?

• 9T + 16T^3

• 9T + 8T^3

• 9T + (8/3)T^3

• 9T + (16/3)T^3

What is the value of the integral when T is 2?

• 182

• 364

• 128

• 256

What is the final answer for the line integral?

• 364√3 / 3

• 256√3 / 3

• 128√3 / 3

• 182√3 / 3