# Definite Integral from 0 to a of Root of a Squared minus x Squared

## Theorem

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

for $a > 0$.

## Proof 1

 $\ds \int_0^a \sqrt {a^2 - x^2} \rd x$ $=$ $\ds \intlimits {\frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a} 0 a$ Primitive of $\sqrt {a^2 - x^2}$ $\ds$ $=$ $\ds \paren {\frac {a \sqrt {a^2 - a^2} } 2 + \frac {a^2} 2 \arcsin \frac a a} - \paren {\frac {0 \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac 0 a}$ $\ds$ $=$ $\ds \frac {a^2} 2 \arcsin 1 - \frac {a^2} 2 \arcsin 0$ removing vanishing terms $\ds$ $=$ $\ds \frac {a^2} 2 \arcsin 1$ Sine of Zero is Zero $\ds$ $=$ $\ds \frac \pi 2 \frac {a^2} 2$ Sine of Right Angle

Hence the result.

$\blacksquare$

## Proof 2

 $\ds \int_0^a \sqrt {a^2 - x^2} \rd x$ $=$ $\ds \frac {a^{1 + \frac 2 2} } 2 \frac {\map \Gamma {\frac 1 2} \map \Gamma {1 + \frac 1 2} } {\map \Gamma {\frac 1 2 + \frac 1 2 + 1} }$ 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

$\blacksquare$