高専の数学解説

第8章 8.11

(1)

∬DxydxdyD:x2+y2≤a2x≥0y≥0\iint_D xydxdy \qquad D: x^2+y^2 \leq a^2 \quad x \geq 0 \quad y \geq 0

∫0a∫0a2−x2xy dydx=∫0a[12xy2]0a2−x2dx=∫0a12x(a2−x2)dx=[14a2x2−18x4]0a=a48\int_0^a \int_0^{\sqrt{a^2-x^2}} xy\,dydx = \int_0^a \left[\frac{1}{2}xy^2\right]_0^{\sqrt{a^2-x^2}} dx = \int_0^a \frac{1}{2}x(a^2-x^2)dx = \left[\frac{1}{4}a^2x^2-\frac{1}{8}x^4\right]_0^a = \frac{a^4}{8}

(2)

∬DxydxdyD:x2+y2≤2xy≥0\iint_D xydxdy \qquad D: x^2+y^2 \leq 2x \quad y \geq 0

x=rcos⁡θx = r\cos\theta y=rsin⁡θy = r\sin\theta とおくと 0≤r≤2cos⁡θ0 \leq r \leq 2\cos\theta 0≤θ≤π20 \leq \theta \leq \dfrac{\pi}{2}

∫0π2∫02cos⁡θr3cos⁡θ⋅sin⁡θ drdθ=∫0π2[14r4cos⁡θ⋅sin⁡θ]02cos⁡θdθ=∫0π24cos⁡5θ⋅sin⁡θ dθ\int_0^{\frac{\pi}{2}} \int_0^{2\cos\theta} r^3\cos\theta \cdot \sin\theta \, drd\theta = \int_0^{\frac{\pi}{2}} \left[\frac{1}{4}r^4\cos\theta\cdot\sin\theta\right]_0^{2\cos\theta} d\theta = \int_0^{\frac{\pi}{2}} 4\cos^5\theta\cdot\sin\theta \, d\theta

cos⁡θ=t\cos\theta = t とおくと −sin⁡θdθ=dt-\sin\theta d\theta = dt θ:0→π2\theta: 0 \to \dfrac{\pi}{2} t:1→0t: 1 \to 0 これから

=∫014t5dt=[23t6]01=23= \int_0^1 4t^5 dt = \left[\frac{2}{3}t^6\right]_0^1 = \frac{2}{3}

(3)

∬Dex2+y2dxdy\iint_D e^{x^2+y^2} dxdy D:x2+y2≤a2D: x^2+y^2 \leq a^2 x=rcos⁡θx = r\cos\theta y=rsin⁡θy = r\sin\theta とおくと 0≤r≤a0 \leq r \leq a 0≤θ≤2π0 \leq \theta \leq 2\pi

∫02π∫0aer2r drdθ=∫02π[12er2]0adθ=∫02π12(ea2−1)dθ=π(ea2−1)\int_0^{2\pi}\int_0^a e^{r^2} r \, drd\theta = \int_0^{2\pi}\left[\frac{1}{2}e^{r^2}\right]_0^a d\theta = \int_0^{2\pi}\frac{1}{2}(e^{a^2}-1)d\theta = \pi(e^{a^2}-1)

(4)

∬DxdxdyD:a2≤x2+y2≤2ayx≥0y≥0\iint_D xdxdy \qquad D: a^2 \leq x^2+y^2 \leq 2ay \quad x \geq 0 \quad y \geq 0

x=rcos⁡θx = r\cos\theta y=rsin⁡θy = r\sin\theta とおくと a≤r≤2asin⁡θa \leq r \leq 2a\sin\theta π6≤θ≤π2\dfrac{\pi}{6} \leq \theta \leq \dfrac{\pi}{2}

∫π6π2∫a2asin⁡θr2cos⁡θ drdθ=∫π6π2[13r3cos⁡θ]a2asin⁡θdθ=∫π6π283a3sin⁡3θ⋅cos⁡θ−13a3cos⁡θ dθ\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \int_a^{2a\sin\theta} r^2\cos\theta \, drd\theta = \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \left[\frac{1}{3}r^3\cos\theta\right]_a^{2a\sin\theta} d\theta = \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{8}{3}a^3\sin^3\theta\cdot\cos\theta - \frac{1}{3}a^3\cos\theta \, d\theta

sin⁡θ=t\sin\theta = t とおくと cos⁡θdθ=dt\cos\theta d\theta = dt θ:π6→π2\theta: \dfrac{\pi}{6}\to\dfrac{\pi}{2} t:12→1t: \dfrac{1}{2}\to 1 これから

=∫12183a3t3dt−∫π6π213a3cos⁡θ dθ=83a3[14t4]121−13a3[sin⁡θ]π6π2=1124a3= \int_{\frac{1}{2}}^1 \frac{8}{3}a^3t^3 dt - \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{1}{3}a^3\cos\theta \, d\theta = \frac{8}{3}a^3\left[\frac{1}{4}t^4\right]_{\frac{1}{2}}^1 - \frac{1}{3}a^3\left[\sin\theta\right]_{\frac{\pi}{6}}^{\frac{\pi}{2}} = \frac{11}{24}a^3

解説: r31bn1z