Question

Find the exact length of the curve.x=6+3 t², \quad y=8+2 t³, 0 ≤ t ≤ 2

Answer 1

To find the exact length of the curve defined by the parametric equations x=6+3t² and y=8+2t³, we use the arc length formula that involves the derivatives of x and y with respect to t, and then evaluate the integral from t = 0 to t = 2.

Step-by-step explanation:

To find the exact length of the curve given by the parametric equations x=6+3t² and y=8+2t³ from t = 0 to t = 2, we use the arc length formula for parametric curves:

1. Calculate the derivatives dx/dt and dy/dt.
2. Substitute these into the integral ∫ √((dx/dt)² + (dy/dt)²) dt to find the arc length.

First, we find the derivatives:

dx/dt = 6t, and dy/dt = 6t².

Next, we substitute these into the arc length formula:

Length = ∫ √((6t)² + (6t²)²) dt from t=0 to t=2

Evaluate the integral to get the exact length of the curve.