Understanding Arc Length in Parametric Form

This video tutorial explains how to determine the arc length of curves given in parametric form. It introduces the integral formula for arc length and demonstrates its application through two examples. The first example calculates the arc length of a circle, while the second example uses u-substitution to find the arc length of a more complex curve. The video concludes with a graphical representation of the results.

What is the formula for arc length in parametric form?

• Integral of (dx/dt) + (dy/dt) with respect to t

• Integral of the square root of (x^2 + y^2) with respect to t

• Integral of (x + y) with respect to t

• Integral of the square root of (dx/dt)^2 + (dy/dt)^2 with respect to t

In the first example, what are the parametric equations used to describe the circle?

• x = 2 sin(t), y = 2 cos(t)

• x = 2 cos(t), y = 2 sin(t)

• x = cos(t), y = sin(t)

• x = 2t, y = 2t

What is the arc length of the circle from the first example?

• 2π

• 8π

• 4π

• π

How does the arc length formula relate to the circumference of a circle?

• Arc length is unrelated to circumference

• Arc length is twice the circumference

• Arc length is the same as the circumference

• Arc length is half the circumference

What technique is used to evaluate the integral in the second example?

• Integration by parts

• Partial fraction decomposition

• U-substitution

• Trigonometric substitution

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

• 2

• 2t

• t^2

• t

What is the approximate arc length found in the second example?

• 6.28

• 2.72

• 3.14

• 4.88

What are the coordinates of the point when t = 0 in the second example?

• (-1, -2)

• (0, 0)

• (2, 3)

• (1, 2)

What are the coordinates of the point when t = 2 in the second example?

• (3, 2/3)

• (2, 1)

• (1, 3)

• (0, 2)

What does the arc length represent in the second example?

• The height of the curve

• The distance between two points on the curve

• The distance between two points on a straight line

• The width of the curve