Calculus Concepts and Techniques

This video tutorial explains how to find the arc length of a parametric function. It begins with an introduction to the concept and the necessary conditions for calculating arc length. The video then walks through two example problems, demonstrating the use of the arc length formula and integration techniques, such as u-substitution, to solve for the arc length of given parametric equations.

What is a necessary condition for the arc length formula to be applicable to a parametric function?

• The curve must be closed.

• The curve can only be transverse once as t increases.

• The function must be periodic.

• The function must be linear.

In the first example, what is the derivative of x with respect to t?

• 12t^2

• 6t

• 6t^2

• 12t

What technique is recommended for integrating the expression in the first example?

• Integration by parts

• Partial fraction decomposition

• Trigonometric substitution

• U-substitution

In the first example, what is the final value of the arc length?

• 104

• 42

• 26

• 81

What is the value of 9 raised to the power of 3/2?

• 18

• 27

• 81

• 9

In the second example, what is the expression for dy/dt?

• 9t^2

• 9 - 9t^2

• 18t

• 9t - 3t^3

What is the greatest common factor factored out in the second example?

• 81

• 27

• 9

• 18

What is the result of factoring the expression t^4 + 2t^2 + 1 in the second example?

• (t + 1)^2

• (t + 1)(t - 1)

• (t^2 + 1)(t^2 + 1)

• (t^2 - 1)(t^2 + 1)

What is the antiderivative of t^2 with respect to t?

• t^3/3

• t^2/2

• 3t^2

• t^3

What is the final arc length calculated in the second example?

• 104

• 27

• 42

• 81