What does the arc length formula give for the length of the line r(t)=<t,t,t> for 0 ≤ t ≤ 1?
The arc length formula gives the length of a curve in three-dimensional space. In this case, we have the parametric equation of a line defined by the vector function r(t) = <t, t, t> with the parameter t ranging from 0 to 1.
To find the length of this line segment, we can use the arc length formula, which is given by:
L = ∫ [a to b] √(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 dt
where a and b represent the parameter values that define the range of t. In our case, a = 0 and b = 1.
Let's calculate the arc length step by step:
1. Calculate the derivatives of x(t), y(t), and z(t) with respect to t.
dx/dt = d/dt(t) = 1
dy/dt = d/dt(t) = 1
dz/dt = d/dt(t) = 1
2. Substitute the derivative values into the arc length formula.
L = ∫ [0 to 1] √(1)^2 + (1)^2 + (1)^2 dt
= ∫ [0 to 1] √3 dt
3. Evaluate the integral.
L = [t√3] [0 to 1]
= √3 - 0√3
= √3
Therefore, the length of the line segment represented by r(t) = <t, t, t> for 0 ≤ t ≤ 1 is √3.