Final answer:
To calculate the arclength of the curve y = 3x³/² between x = 1 and x = 3, find the derivative, square it, plug into the arclength formula, and integrate from 1 to 3. This process might require substitution or numerical methods to evaluate the integral.
Step-by-step explanation:
To find the arclength of the curve y = 3x³/² between x = 1 and x = 3, we use the formula for arclength in Cartesian coordinates:
S = ∫₁³ √(1 + (dy/dx)²) dx
First, we need to find the derivative dy/dx of the function y = 3x³/². The derivative is dy/dx = 9x¹/². Then, we square the derivative:
(dy/dx)² = (9x¹/²)² = 81x
Next, we plug this into the arclength formula and integrate between the limits of 1 and 3:
S = ∫₁³ √(1 + 81x) dx
The integration can be complex and typically requires substitution or numerical methods. The exact process for integration and finding the numerical value can vary depending on the preferences of the calculation method or the available tools.
Once the integral is computed, the result is the length of the arc for the given portion of the curve.