Final answer:
The length of a curve represented by parametric equations can be found using the arc length formula. In this case, the length of the curve represented by r = i t^2 j t^3 can be found by integrating the magnitude of the derivative of the vector function with respect to t.
Step-by-step explanation:
The length of a curve represented by parametric equations can be found using the arc length formula. Given the curve r = i t^2 j t^3, we can find the length by integrating the magnitude of the derivative of the vector function with respect to t.
In this case, the derivative of r with respect to t is d r / d t = 2 i t j t^3 + 3 i t^2 j t^2 = (2 t + 3 t^2) i t^3 j.
To find the length of the curve, we integrate the magnitude of the derivative over the interval of t. The magnitude of the derivative is sqrt((2 t + 3 t^2)^2 + (t^3)^2) = sqrt(4 t^2 + 12 t^3 + 9 t^4 + t^6). We integrate this expression over the interval of t to find the length of the curve.