3.1

305
Lasketaan eri puolien pinta-alojen raja-arvot

Koska funktio ei ole määritelty kohdassa x ≤ 0, pinta-ala saadaan seuraavasti

$\lim_{t\rightarrow0}\int_t^1\frac{1}{4\sqrt[]{x}}dx=\lim_{t\rightarrow0}\int_t^1\frac{1}{x^{\frac{1}{4}}}dx=\lim_{t\rightarrow0}\int_t^1x^{-\frac{1}{4}}dx=\lim_{t\rightarrow0}\bigg/_{\!\!\!\!\!t}^1\frac{4}{3}x^{\frac{3}{4}}=\lim_{t\rightarrow0}\left(\frac{4}{3}-\frac{4}{3}t^{\frac{4}{3}}\right)=\frac{4}{3}-0=\frac{4}{3}$

$\lim_{i\rightarrow\infty}\int_1^i\frac{1}{\sqrt[4]{x}}dx=\lim_{i\rightarrow\infty}\int_1^i\frac{1}{^{x^{\frac{1}{4}}}}dx=\lim_{i\rightarrow\infty}\int_1^ix^{-\frac{1}{4}}dx=\lim_{i\rightarrow\infty}\bigg/_{\!\!\!\!\!1}^i\frac{4}{3}x^{\frac{3}{4}}=\lim_{i\rightarrow\infty}\left(\ \frac{4}{3}i^{\frac{3}{4}}-\frac{4}{3}\right)=\infty$

Ensimmäinen pinta-ala on äärellinen, toinen on ääretön

306

$V=\pi\int_0^{\ t}\left(\frac{1}{\sqrt[]{e^x}}\right)^2dx=\pi\int_0^{\ t}\frac{1}{e^x}dx=\pi\int_0^{\ t}\frac{1}{e^x}dx=\pi\int_0^{\ t}e^{-x}dx=\pi\bigg/_{\!\!\!\!\!0}^t-e^{-x}$
$=\pi\left(-e^{-t}-\left(-e^0\right)\right)=\pi\left(1-\frac{1}{e^t}\right)$

$\pi\left(1-\frac{1}{e^t}\right)\rightarrow\pi\left(1-0\right)=\pi$

310

Kuvaaja on x-akselin alapuolella, joten pinta-ala on integraalin vastaluku

$-\int_0^1\frac{x^2-1}{\sqrt[]{x}}dx=\lim_{t\rightarrow0+}\int_t^1\frac{x^2-1}{\sqrt[]{x}}dx$

$=-\lim_{t\rightarrow0+}\int_t^1\frac{x^2}{\sqrt[]{x}}-\frac{1}{\sqrt[]{x}}dx$

$=-\lim_{t\rightarrow0+}\int_t^1\left(x^{\frac{3}{2}}-x^{-\frac{1}{2}}\right)dx$

$=-\lim_{t\rightarrow0+}\bigg/_{\!\!\!\!\!t}^1\left(\frac{2}{5}x^{\frac{5}{2}}-2x^{\frac{1}{2}}\right)dx$

$=-\lim_{t\rightarrow0+}\bigg/_{\!\!\!\!\!t}^1\left(\frac{2}{5}x^2\sqrt[]{x}-2\sqrt[]{x}\right)dx$

$=-\lim_{t\rightarrow0+}\left(\frac{2}{5}\cdot1-2\cdot1-\left(\frac{2}{5}t^2\sqrt[]{t}-2\sqrt[]{t}\right)\right)$
$=-\left(\frac{2}{5}-2-0\right)$
$=1\frac{3}{5}$

$\int_{-1}^1\frac{1}{\sqrt[3]{x^2}}dx=\int_{-1}^0\frac{1}{\sqrt[3]{x^2}}dx+\int_0^1\frac{1}{\sqrt[3]{x^2}}dx$

$=\lim_{t\rightarrow0-}\int_{-1}^tx^{-\frac{2}{3}}dx+\lim_{t\rightarrow0+}\int_t^1x^{-\frac{2}{3}}dx$

$=\lim_{t\rightarrow0-}\bigg/_{\!\!\!\!\!{-1}}^t3x^{\frac{1}{3}}+\lim_{t\rightarrow0+}\bigg/_{\!\!\!\!\!t}^13x^{\frac{1}{3}}$

$=\lim_{t\rightarrow0-}\bigg/_{\!\!\!\!\!{-1}}^t3\sqrt[3]{x}+\lim_{t\rightarrow0+}\bigg/_{\!\!\!\!\!t}^13\sqrt[3]{x}$

$=\lim_{t\rightarrow0-}\left(3\sqrt[3]{t}-3\sqrt[3]{1}\right)+\lim_{t\rightarrow0+}\left(3\sqrt[3]{1}-3\sqrt[3]{t}\right)dx$

$=3\cdot0-3\cdot\left(-1\right)+3\cdot1-3\cdot0$

$=6$

313

316

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