Question

Hayar la superficie limitada por la curva ay^2=x^3; y=0 i y=a

Answer 1

                             $f(x)=\begin{cases} -\sqrt{\dfrac{x^3}{a}}\\ \\ \sqrt{\dfrac{x^3}{a}} \end{cases}\\ \\ \displaystyle \dfrac{1}{2}A=\int_{0}^{a}\sqrt{\dfrac{x^3}{a}}~dx\\ \\ \\ A=2\sqrt{a}\int_{0}^{a}x^{3/2}~dx\\ \\ \\ A=2\sqrt{a}\cdot \left.\dfrac{2}{5}x^{5/2}\right|_{0}^ a\\ \\ \\ \boxed{A=\dfrac{4}{5}a^3}$