Area of an Astroid: x^2/3+y^2/3=a^2/3

Описание: To find the area enclosed by this astroid, you can use calculus and integrate. The equation of the astroid is:

x^(2/3) + y^(2/3) = a^(2/3)

To find the area enclosed by this curve, you can integrate it from one side to the other. Since this is a symmetric curve, you can consider just one part of it and then multiply the result by 4 to account for all four lobes of the astroid.

Let's consider the upper-right lobe (quadrant I) of the astroid, which can be defined by the following parametric equations:

x(t) = a * cos^3(t)
y(t) = a * sin^3(t)

where t ranges from 0 to π/2 (one-quarter of a full revolution). Now, we can find the area of this lobe by integrating the differential area element:

dA = x dy = (a * cos^3(t)) * (3a * sin^2(t) * cos(t)) dt

To find the total area of this lobe, integrate dA with respect to t from 0 to π/2:

A = 4 * ∫[0, π/2] (3a^2 * cos^4(t) * sin^2(t)) dt

You can simplify this integral using trigonometric identities and then calculate the result.

Once you have the result for the area of one lobe, multiply it by 4 to get the total area enclosed by the astroid. The exact integral may not have a simple closed-form expression and might require numerical methods or a computer algebra system to evaluate.
   / @drbmbkrushna_mvgr_maths  
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