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Calculate the length of the astroid of x²/³+y² = 5.

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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.

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