Definite Integral from 0 to a of Root of a Squared minus x Squared/Proof 2

Theorem

$\ds \int_0^a \sqrt {a^2 - x^2} \rd x = \frac {\pi a^2} 4$

for $a > 0$.

Proof

\(\ds \int_0^a \sqrt {a^2 - x^2} \rd x\)

\(=\)

\(\ds \frac {a^{1 + \frac 2 2} } 2 \frac {\map \Gamma {\frac 1 2} \map \Gamma {1 + \frac 1 2} } {\map \Gamma {\frac 1 2 + \frac 1 2 + 1} }\)

Definite Integral from 0 to a of $x^m \paren {a^n - x^n}^p$

\(\ds \)

\(=\)

\(\ds \frac {a^2} 2 \frac {\map \Gamma {\frac 3 2} \map \Gamma {\frac 1 2} } {\map \Gamma 2}\)

\(\ds \)

\(=\)

\(\ds \frac {a^2} {2 \times 1!} \paren {\frac 1 2} \paren {\map \Gamma {\frac 1 2} }^2\)

Gamma Difference Equation, Gamma Function Extends Factorial

\(\ds \)

\(=\)

\(\ds \frac {\pi a^2} 4\)

Gamma Function of One Half

$\blacksquare$