Final answer:
To calculate the length of the astroid, we need to find the interval of the astroid curve from a minimum value of x to a maximum value of x. Let's start by solving the equation x²/³ + y² = 5 for y in terms of x. Next, we take the derivative of y with respect to x, and then integrate the square root of 1 + (dy/dx)² to find the length of the astroid.
Step-by-step explanation:
To calculate the length of the astroid, we need to find the interval of the astroid curve from a minimum value of x to a maximum value of x. Let's start by solving the equation x²/³ + y² = 5 for y in terms of x. We can rewrite the equation as y = ±(5 - x²/³)². Next, we take the derivative of y with respect to x, which is dy/dx = ±4x(5 - x²/³)-²/³. To find the interval of the curve, we need to integrate the square root of 1 + (dy/dx)² with respect to x from the minimum value of x to the maximum value of x. Once we integrate and simplify the expression, we can calculate the length of the astroid.