The area of the astroid is [tex]\frac{3\pi\cdot a^{2}}{8}[/tex] square units.
The expression of the astroid is also equivalent to following parametric formulae:
[tex]x(t) = a\cdot \cos^{3} t[/tex] (1)
[tex]y(t) = a\cdot \sin ^{3}t[/tex] (2)
Based on the fact that the astroid has two axes of symmetry and each axis of symmetry is perpendicular to the other, we can determine the total area by multiplying the area between [tex]0[/tex] and [tex]\frac{\pi}{2}[/tex] four times:
[tex]A = 4\cdot \int\limits^{\frac{\pi}{2} }_{0} {x(t)\cdot \dot y(t)} \, dt[/tex] (3)
If we know that [tex]x(t) = a\cdot \cos^{3} t[/tex], [tex]y(t) = a\cdot \sin ^{3}t[/tex] and [tex]\dot y = 3\cdot a\cdot \sin^{2}t\cdot \cos t[/tex], then we have the following integral:
[tex]A = 12\cdot a^{2}\int\limits^{\frac{\pi}{2} }_{0 } {\sin^{2}t\cdot \cos^{4}t} \, dt[/tex] (3b)
This area is equivalent to the following Beta and Gamma formulae:
[tex]A = 6\cdot a^{2}\cdot B\left(\frac{3}{2}, \frac{5}{2} \right) = \frac{6\cdot a^{2}\cdot \Gamma\left(\frac{3}{2} \right)\cdot \Gamma\left(\frac{5}{2} \right)}{\Gamma(4)}[/tex]
By applying Gamma function properties we get the following result:
[tex]A = \frac{6\cdot a^{2}\cdot \left(\frac{3}{8} \right)\left[\Gamma\left(\frac{1}{2} )\right]^{2}}{3!}[/tex]
[tex]A = \frac{3\pi\cdot a^{2}}{8}[/tex]
The area of the astroid is [tex]\frac{3\pi\cdot a^{2}}{8}[/tex] square units. [tex]\blacksquare[/tex]
To learn more on definite integrals, we kindly invite to check this verified question: https://brainly.com/question/22655212
