【微分積分】4-7-2 曲線の長さ｜問題集

求める曲線\(L\)は
\(\displaystyle L=\int_0^2\sqrt{\left(\frac{dx}{dx}\right)^2+\left(\frac{dy}{dx}\right)^2}dx\)
\(\displaystyle =\int_0^2\sqrt{1+2^2}dx\)
\(\displaystyle =[\sqrt{5}x]_0^2\)
\(\displaystyle =2\sqrt{5}\)

求める曲線\(L\)は
\(\displaystyle L=\int_0^{44}\sqrt{\left(\frac{dx}{dx}\right)^2+\left(\frac{dy}{dx}\right)^2}dx\)
\(\displaystyle =\int_0^{44}\sqrt{1+\left(\frac{3x^{\frac{1}{2}}}{2}\right)^2}dx\)
\(\displaystyle =\int_1^{100}\sqrt{t}・\frac{4}{9}dt\)
\(\displaystyle =\frac{4}{9}・\frac{2}{3}\left[t^{\frac{3}{2}}\right]_1^{100}\)
\(\displaystyle =296\)

求める曲線\(L\)は
\(\displaystyle L=\int_0^\sqrt{3}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt\)
\(\displaystyle =\int_0^\sqrt{3}\sqrt{(2t)^2+2^2}dt\)
\(\displaystyle =2\int_0^\sqrt{3}\sqrt{t^2+1}dt\)
\(\displaystyle =\left[t\sqrt{t^2+1}+\log\left|t+\sqrt{t^2+1}\right|\right]_0^\sqrt{3}\)
\(\displaystyle =2\sqrt{3}+\log|2+\sqrt{3}|\)

求める曲線\(L\)は
\(\displaystyle L=\int_0^{4\pi}\sqrt{\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2}d\theta\)
\(\displaystyle =\int_0^{4\pi}\sqrt{2e^{2\theta}}d\theta\)
\(\displaystyle =\sqrt{2}\int_0^{4\pi}e^\theta d\theta\)
\(\displaystyle =\sqrt{2}[e^\theta]_0^{4\pi}\)
\(\displaystyle =\sqrt{2}(e^{4\pi}-1)\)

求める曲線\(L\)は
\(\displaystyle L=4\int_0^{\frac{\pi}{2}}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt\)
\(\displaystyle =4\int_0^{\frac{\pi}{2}}\sqrt{(-3\cos^2t\sin t)^2+(3\sin^2t\cos t)^2}dt\)
\(\displaystyle =12\int_0^{\frac{\pi}{2}}\cos t\sin tdt\)
\(\displaystyle =12\left[\frac{\sin^2t}{2}\right]_0^{\frac{\pi}{2}}\)
\(\displaystyle =6\)

求める曲線\(L\)は
\(\displaystyle L=\int_0^1\sqrt{(2t)^2+\{-2(1-t)\}^2}dt\)
\(\displaystyle =2\int_0^1\sqrt{t^2+(1-t)^2}dt\)
\(\displaystyle =1+\frac{\sqrt{2}}{2}\log(1+\sqrt{2})\)

求める曲線\(L\)は
\(\displaystyle L=\int_0^{2\pi}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta\)
\(\displaystyle =2\int_0^{\pi}\sqrt{(1+\cos\theta)^2+\sin^2\theta}d\theta\)
\(\displaystyle =4\int_0^{\pi}\cos\frac{\theta}{2}d\theta\)
\(\displaystyle =4・2\left[\sin\frac{\theta}{2}\right]_0^{\pi}\)
\(\displaystyle =8\)