Solution
To solve: \int^{a}_{0}y^{4}\sqrt{a^{2}-y^{2}}dy (Utilising the Beta special function)
Use the substitution y=a\sqrt{t}
This implies t = \frac{y^{2}}{a^{2}}
Change bounds and variable
for y=0, t=0; for y=a, t = a2/a2 = 1
t=\frac{y^{2}}{a^{2}}\rightarrow \frac{a^{2}}{2y}dt=dy...
That did it thanks. Great skills! (I want them)
I've still got something wrong.
Subbing y=a\sqrt{t} into integrand:
a^{4}t^{2}\sqrt{a^{2}(1-t)}
Changing bounds and var (wrongly?):
t=\frac{y^{2}}{a^{2}}\rightarrow \frac{a^{2}}{2y}dt=dy...
Homework Statement
I have this incomplete Beta function question I need to solve using the Beta function.
\int^{a}_{0}y^{4}\sqrt{a^{2}-y^{2}}dy
Homework Equations
Is there an obvious substitution which will help convert to a variant of Beta?
Beta function and variants are in Beta_function...