第8章 8.4

(1)

$\int_0^{\pi} \int_0^{a\cos\theta} r\sin\theta \, dr d\theta = \int_0^{\pi} \left[\frac{1}{2} r^2 \sin\theta\right]_0^{a\cos\theta} d\theta = \int_0^{\pi} \frac{1}{2} a^2 \cos^2\theta \cdot \sin\theta d\theta$

$\cos\theta = t$ とおくと $-\sin\theta d\theta = dt$ $\theta : 0 \to \pi$ $t : 1 \to -1$ これから

$= \int_{-1}^{1} \frac{1}{2} a^2 t^2 dt = \left[\frac{1}{6} a^2 t^3\right]_{-1}^{1} = \frac{1}{3} a^2$

(2)

$\int_0^{\frac{\pi}{2}} \int_{a\cos\theta}^{a} r \, dr d\theta = \int_0^{\frac{\pi}{2}} \left[\frac{1}{2} r^2\right]_{a\cos\theta}^{a} d\theta = \int_0^{\frac{\pi}{2}} \frac{1}{2} a^2 (1-\cos^2\theta) d\theta = \int_0^{\frac{\pi}{2}} \frac{1}{2} a^2 \sin^2\theta d\theta$

$= \frac{1}{2} a^2 \int_0^{\frac{\pi}{2}} \frac{1-\cos 2\theta}{2} d\theta = \frac{1}{2} a^2 \left[\frac{1}{2}\theta - \frac{1}{4}\sin 2\theta\right]_0^{\frac{\pi}{2}} = \frac{\pi}{8} a^2$

(3)

$\iint_D r^2 \sin\theta \, dr d\theta \qquad D : 0 \le r \le 2a\cos\theta \quad 0 \le \theta \le \frac{\pi}{2}$

$\int_0^{\frac{\pi}{2}} \int_0^{2a\cos\theta} r^2 \sin\theta \, dr d\theta = \int_0^{\frac{\pi}{2}} \left[\frac{1}{3} r^3 \sin\theta\right]_0^{2a\cos\theta} d\theta = \int_0^{\frac{\pi}{2}} \frac{8}{3} a^3 \cos^3\theta \cdot \sin\theta d\theta$

$\cos\theta = t$ とおくと $-\sin\theta d\theta = dt$ $\theta : 0 \to \dfrac{\pi}{2}$ $t : 1 \to 0$ これから

$= \int_0^{1} \frac{8}{3} a^3 t^3 dt = \left[\frac{2}{3} a^3 t^4\right]_0^{1} = \frac{2}{3} a^3$