Final answer:
To find the length of the curve r(t) = t * e⁻t, we can use the arc length formula. The arc length formula for a curve in parametric form is given by: As = ∫ √(r'(t)² + r(t)²) dt. In this case, the derivative of r(t) is r'(t) = (e⁻t - t * e⁻t). To calculate the integral, we can simplify the expression inside the square root and then integrate.
Step-by-step explanation:
To find the length of the curve r(t) = t * e⁻t, we can use the arc length formula. The arc length formula for a curve in parametric form is given by:
As = ∫ √(r'(t)² + r(t)²) dt
In this case, the derivative of r(t) is r'(t) = (e⁻t - t * e⁻t). To calculate the integral, we can simplify the expression inside the square root and then integrate:
√(r'(t)² + r(t)²) = √((e⁻t - t * e⁻t)² + (t * e⁻t)²)
After expanding and simplifying, we get:
√(e⁻²t + 2te⁻²t + t²e⁻²t + t²e⁻²t) = √(2te⁻²t + 2t²e⁻²t)
Finally, we can integrate this expression with respect to t to find the arc length of the curve.