find the arc length of the curve r(t)= <6sqrt(2),e^t,e^(-6t)>
To find the arc length of a curve, we use the formula:
L = ∫ ||r'(t)|| dt
where r(t) is the vector-valued function representing the curve, r'(t) is the derivative of r(t), and ||r'(t)|| is the magnitude (or length) of the derivative vector.
In this case, we have the vector-valued function r(t) = <6√2, e^t, e^(-6t)>. To find the derivative r'(t), we differentiate each component of r(t) with respect to t:
r'(t) = <d/dt(6√2), d/dt(e^t), d/dt(e^(-6t))>
= <0, e^t, -6e^(-6t)>
To find the magnitude ||r'(t)||, we use the formula:
||r'(t)|| = √(0^2 + (e^t)^2 + (-6e^(-6t))^2)
= √(e^(2t) + 36e^(-12t))
Now we have all the information we need to calculate the arc length.
L = ∫ √(e^(2t) + 36e^(-12t)) dt
However, calculating this integral exactly may be quite challenging. If you prefer an exact answer, you can use numerical methods or computer software to approximate the integral. Alternatively, if an approximation is acceptable, you can provide a specific range of t values to find the approximate arc length over that interval.