定积分demo

计算定积分

\int_{0}^{a} = \sqrt{a^{2} - x^{2}} d x

a > 0

)

解：令 $x = a s i n t$ ,即 $d x = d (a s i n t) = a c o s t d t$ ，且当 $x$ 从 $0$ 变化到 $a$ 时， $t$ 从 $0$ 变化到 $\frac{π}{2}$ ，从而

$\int_{0}^{a} = \sqrt{a^{2} - x^{2}} d x = \int_{0}^{\frac{π}{2}} a c o s t \times a c o s t d t = a^{2} \int_{0}^{\frac{π}{2}} {c o s}^{2} t d t$

$= a^{2} \int_{0}^{\frac{π}{2}} \frac{1 + c o s 2 t}{2} d t = \frac{a^{2}}{2} (t + \frac{s i n 2 t}{2}) |_{0}^{\frac{π}{2}} = \frac{1}{4} π a^{2}$