## Vad är en rotationsvolym? När ett område i planet **roterar** kring en axel skapas en tredimensionell kropp. Volymen av denna kropp kallas **rotationsvolym**. ```tikz \begin{document} \begin{tikzpicture}[scale=1.1] % Vänster: 2D-kurvan \begin{scope}[xshift=0cm] \draw[->, thick] (-0.3,0) -- (3.5,0) node[right] {$x$}; \draw[->, thick] (0,-0.3) -- (0,2.5) node[above] {$y$}; \fill[blue!20] (0.5,0) -- plot[domain=0.5:3, samples=50] (\x, {0.4*sqrt(\x)+0.3}) -- (3,0) -- cycle; \draw[blue, very thick, domain=0.3:3.2, samples=100] plot (\x, {0.4*sqrt(\x)+0.3}); \node[blue] at (2.5, 1.8) {$y = f(x)$}; \node at (1.7, -0.8) {\textbf{2D: Område}}; \end{scope} % Pil \draw[->, ultra thick, red!70!black] (4, 1) -- (5.5, 1); \node[red!70!black] at (4.75, 1.5) {\small rotation}; % Höger: 3D-kroppen (förenklad vy) \begin{scope}[xshift=6.5cm] \draw[->, thick] (-0.3,0) -- (3.5,0) node[right] {$x$}; \draw[->, thick] (0,-2) -- (0,2.3) node[above] {$y$}; % Övre kontur \draw[blue, very thick, domain=0.5:3, samples=100] plot (\x, {0.4*sqrt(\x)+0.3}); % Undre kontur (spegling) \draw[blue, very thick, domain=0.5:3, samples=100] plot (\x, {-0.4*sqrt(\x)-0.3}); % Ellipser för 3D-effekt \draw[blue, thick] (0.5, 0) ellipse (0.08 and 0.5); \draw[blue, thick] (3, 0) ellipse (0.12 and 0.99); \draw[blue, thick, dashed] (1.75, 0) ellipse (0.1 and 0.83); % Skuggning \fill[blue!10, opacity=0.5] (0.5, 0.5) -- plot[domain=0.5:3, samples=50] (\x, {0.4*sqrt(\x)+0.3}) -- (3, 0.99) arc (90:-90:0.12 and 0.99) -- plot[domain=3:0.5, samples=50] (\x, {-0.4*sqrt(\x)-0.3}) -- (0.5, -0.5) arc (-90:90:0.08 and 0.5); \node at (1.7, -2.5) {\textbf{3D: Rotationskropp}}; \end{scope} \end{tikzpicture} \end{document} ``` --- ## De två metoderna ### Skivmetoden Tänk dig att kroppen byggs upp av tunna cirkulära **skivor**. ```tikz \begin{document} \begin{tikzpicture}[scale=1.3] % Axlar \draw[->, thick, gray] (-0.5,0) -- (5.5,0) node[right] {$x$}; \draw[->, thick, gray] (0,-2.2) -- (0,2.5) node[above] {$y$}; % Rotationskroppens konturer \draw[blue!70, very thick, domain=0.3:5, samples=100] plot (\x, {0.5*sqrt(\x)+0.2}); \draw[blue!70, very thick, domain=0.3:5, samples=100] plot (\x, {-0.5*sqrt(\x)-0.2}); % Flera skivor \foreach \x/\op in {0.8/0.3, 1.5/0.4, 2.3/0.5, 3.2/0.6, 4.2/0.7} { \pgfmathsetmacro{\r}{0.5*sqrt(\x)+0.2} \fill[red!40, opacity=\op] (\x, 0) ellipse (0.15 and \r); \draw[red!70!black, thick] (\x, 0) ellipse (0.15 and \r); } % En markerad skiva \fill[orange!60] (3.2, 0) ellipse (0.15 and 1.09); \draw[orange!80!black, very thick] (3.2, 0) ellipse (0.15 and 1.09); % Radie \draw[<->, green!50!black, very thick] (3.2, 0) -- (3.2, 1.09); \node[green!50!black] at (3.6, 0.55) {$r = f(x)$}; % Tjocklek \draw[<->, purple, thick] (3.05, -1.5) -- (3.35, -1.5); \node[purple] at (3.2, -1.8) {$dx$}; % Formel \node[draw, fill=white, rounded corners] at (2.5, 2.2) {$dV = \pi r^2 \cdot dx = \pi[f(x)]^2 dx$}; \end{tikzpicture} \end{document} ``` $\boxed{V = \pi \int_a^b [f(x)]^2 , dx}$ --- ### Skalmetoden Tänk dig cylindriska **skal** (rör) som sitter inuti varandra. ```tikz \begin{document} \begin{tikzpicture}[scale=1.2] % Axlar \draw[->, thick, gray] (-0.5,0) -- (4,0) node[right] {$x$}; \draw[->, thick, gray] (0,-0.5) -- (0,3.5) node[above] {$y$}; % Området som roteras \fill[blue!15] (0,0) -- plot[domain=0:2.8, samples=50] (\x, {-0.25*(\x-1.4)^2 + 2.5}) -- (2.8,0) -- cycle; \draw[blue, very thick, domain=0:3, samples=100] plot (\x, {-0.25*(\x-1.4)^2 + 2.5}); % Cylindriska skal \foreach \r/\col in {0.6/red!30, 1.2/orange!40, 1.8/yellow!50, 2.4/green!30} { \pgfmathsetmacro{\h}{-0.25*(\r-1.4)^2 + 2.5} \fill[\col] (\r-0.08, 0) rectangle (\r+0.08, \h); \draw[black!50, thick] (\r-0.08, 0) -- (\r-0.08, \h); \draw[black!50, thick] (\r+0.08, 0) -- (\r+0.08, \h); \draw[black!50, thick] (\r-0.08, \h) -- (\r+0.08, \h); } % Ett markerat skal \pgfmathsetmacro{\rmark}{1.8} \pgfmathsetmacro{\hmark}{-0.25*(\rmark-1.4)^2 + 2.5} \fill[red!50] (\rmark-0.1, 0) rectangle (\rmark+0.1, \hmark); \draw[red!80!black, very thick] (\rmark-0.1, 0) rectangle (\rmark+0.1, \hmark); % Markeringar \draw[<->, green!50!black, very thick] (0, -0.3) -- (\rmark, -0.3); \node[green!50!black] at (0.9, -0.6) {$x$}; \draw[<->, purple, very thick] (2.1, 0) -- (2.1, \hmark); \node[purple] at (2.5, 1.2) {$f(x)$}; \draw[<->, orange, thick] (\rmark-0.1, 2.7) -- (\rmark+0.1, 2.7); \node[orange] at (\rmark, 3) {$dx$}; % Rotationsaxel \draw[dashed, thick, gray] (0, -0.3) -- (0, 3.3); \node[gray] at (-0.4, 3) {\small y-axel}; % Formel \node[draw, fill=white, rounded corners] at (3.5, 2.8) {$dV = 2\pi x \cdot f(x) \cdot dx$}; \end{tikzpicture} \end{document} ``` $\boxed{V = 2\pi \int_a^b x \cdot f(x) , dx}$ --- ## Klassiska exempel ### 1. Sfären Rotera halvcirkeln $y = \sqrt{R^2 - x^2}$ kring x-axeln. ```tikz \begin{document} \begin{tikzpicture}[scale=1.4] % Axlar \draw[->, thick, gray] (-2.8,0) -- (2.8,0) node[right] {$x$}; \draw[->, thick, gray] (0,-2.3) -- (0,2.5) node[above] {$y$}; % Sfären (cirkel med skuggning) \shade[ball color=blue!40!white, opacity=0.8] (0,0) circle (2); % Konturlinjer \draw[blue!70!black, very thick] (0,0) circle (2); \draw[blue!50, thick, dashed] (0,0) ellipse (2 and 0.5); % Radie \draw[red, very thick, ->] (0,0) -- (1.41, 1.41); \node[red] at (1, 0.5) {$R$}; % Gränser \node at (-2, -0.35) {$-R$}; \node at (2, -0.35) {$R$}; % Formel \node[draw, fill=white, rounded corners] at (0, -3) {$V = \dfrac{4\pi R^3}{3}$}; \end{tikzpicture} \end{document} ``` **Beräkning:** $V = \pi \int_{-R}^{R} (R^2 - x^2) , dx = \pi \left[R^2 x - \frac{x^3}{3}\right]_{-R}^{R} = \frac{4\pi R^3}{3}$ --- ### 2. Konen Rotera linjen $y = \frac{r}{h}x$ kring x-axeln. ```tikz \begin{document} \begin{tikzpicture}[scale=1] % Axlar \draw[->, thick, gray] (-0.5,0) -- (5.5,0) node[right] {$x$}; \draw[->, thick, gray] (0,-2.5) -- (0,2.8) node[above] {$y$}; % Konens yta \fill[orange!20] (0,0) -- (4.5, 2) -- (4.5, -2) -- cycle; \shade[left color=orange!40, right color=orange!10] (0,0) -- (4.5, 2) -- (4.5, -2) -- cycle; % Konturer \draw[orange!80!black, very thick] (0,0) -- (4.5, 2); \draw[orange!80!black, very thick] (0,0) -- (4.5, -2); % Basen (ellips) \fill[orange!50] (4.5, 0) ellipse (0.3 and 2); \draw[orange!80!black, thick] (4.5, 0) ellipse (0.3 and 2); % Mått \draw[<->, red, very thick] (0, -2.8) -- (4.5, -2.8); \node[red] at (2.25, -3.2) {$h$}; \draw[<->, green!50!black, very thick] (5, 0) -- (5, 2); \node[green!50!black] at (5.4, 1) {$r$}; % Etikett \node[orange!80!black] at (2, 1.5) {$y = \frac{r}{h}x$}; % Formel \node[draw, fill=white, rounded corners] at (2.25, -4.2) {$V = \dfrac{\pi r^2 h}{3}$}; \end{tikzpicture} \end{document} ``` **Beräkning:** $V = \pi \int_0^h \left(\frac{r}{h}x\right)^2 dx = \frac{\pi r^2}{h^2} \cdot \frac{h^3}{3} = \frac{\pi r^2 h}{3}$ --- ### 3. Torusen Rotera en cirkel med radie $r$ kring en axel på avstånd $R$. ```tikz \begin{document} \begin{tikzpicture}[scale=0.9] % Torus illustration (tvärsnitt + 3D-antydan) % Yttre ellips (torusens kontur) \draw[red!70!black, very thick] (0,0) ellipse (4 and 2); % Inre ellips (hålet) \draw[red!70!black, very thick] (0,0) ellipse (2 and 0.8); % Skuggning övre del \fill[red!20, opacity=0.6] (0,0) ellipse (4 and 2); \fill[white] (0,0) ellipse (2 and 0.8); % 3D-effekt: extra ellipser \draw[red!50, thick, dashed] (-3, 0) arc (180:360:0.5 and 1.2); \draw[red!50, thick] (-3, 0) arc (180:0:0.5 and 1.2); \draw[red!50, thick, dashed] (3, 0) arc (180:360:0.5 and 1.2); \draw[red!50, thick] (3, 0) arc (180:0:0.5 and 1.2); % Cirkeln som roteras (till höger) \begin{scope}[xshift=7cm] \draw[->, thick, gray] (-0.5,0) -- (4,0) node[right] {$x$}; \draw[->, thick, gray] (0,-2) -- (0,2) node[above] {$y$}; % Rotationsaxel \draw[dashed, thick, blue] (0, -1.8) -- (0, 1.8); % Cirkeln \draw[red, very thick, fill=red!20] (2.5, 0) circle (1); % Mått \draw[<->, green!50!black, thick] (0, -1.5) -- (2.5, -1.5); \node[green!50!black] at (1.25, -1.8) {$R$}; \draw[<->, purple, thick] (2.5, 0) -- (3.5, 0); \node[purple] at (3.2, 0.3) {$r$}; \node[gray] at (0.4, 1.5) {\small axel}; \end{scope} % Formel \node[draw, fill=white, rounded corners] at (0, -3.5) {$V = 2\pi^2 R r^2$}; \end{tikzpicture} \end{document} ``` --- ## Inspirerande exempel ### 4. Vinglaset Rotera kurvan $y = x^2$ för $0 \leq x \leq 2$ kring y-axeln. ```tikz \begin{document} \begin{tikzpicture}[scale=1] % Axlar \draw[->, thick, gray] (-2.5,0) -- (2.8,0) node[right] {$x$}; \draw[->, thick, gray] (0,-0.5) -- (0,5) node[above] {$y$}; % Parabeln och dess spegling \draw[purple!70, very thick, domain=0:2, samples=100] plot (\x, {\x*\x}); \draw[purple!70, very thick, domain=0:2, samples=100] plot ({-\x}, {\x*\x}); % Fyllning för 3D-effekt \shade[left color=purple!30, right color=purple!10, opacity=0.7] plot[domain=0:2, samples=50] (\x, {\x*\x}) -- plot[domain=2:0, samples=50] ({-\x}, {\x*\x}) -- cycle; % Ellipser för djup \draw[purple!50, thick] (0, 0.5) ellipse (0.7 and 0.15); \draw[purple!50, thick] (0, 2) ellipse (1.41 and 0.25); \draw[purple!70, thick] (0, 4) ellipse (2 and 0.35); % Glas-form antydan \draw[purple!40, thick, dashed] (0, 4) -- (0, 4.5); \draw[purple!40, thick] (-0.3, 4.5) -- (0.3, 4.5); % Mått \node at (2.2, 4) {$y = 4$}; \node at (2.3, 0.3) {$x = 2$}; % Etikett \node[purple!80!black] at (1.8, 2) {$y = x^2$}; \end{tikzpicture} \end{document} ``` **Med skalmetoden:** $V = 2\pi \int_0^2 x \cdot x^2 , dx = 2\pi \int_0^2 x^3 , dx = 2\pi \left[\frac{x^4}{4}\right]_0^2 = \boxed{8\pi}$ --- ### 5. Gabriels horn — _Det omöjliga hornet_ > _Oändlig yta, men ändlig volym — kan fyllas med färg men aldrig målas!_ Rotera $y = \frac{1}{x}$ för $x \geq 1$ kring x-axeln. ```tikz \begin{document} \begin{tikzpicture}[scale=1.1] % Axlar \draw[->, thick, gray] (-0.3,0) -- (7,0) node[right] {$x$}; \draw[->, thick, gray] (0,-1.8) -- (0,2) node[above] {$y$}; % Hornet (övre och undre kontur) \draw[violet!80!black, very thick, domain=1:6.5, samples=200] plot (\x, {1/\x}); \draw[violet!80!black, very thick, domain=1:6.5, samples=200] plot (\x, {-1/\x}); % Skuggning \shade[top color=violet!30, bottom color=violet!10, opacity=0.6] plot[domain=1:6.5, samples=100] (\x, {1/\x}) -- plot[domain=6.5:1, samples=100] (\x, {-1/\x}) -- cycle; % Ellipser \draw[violet!70, thick] (1, 0) ellipse (0.1 and 1); \draw[violet!50, thick] (2, 0) ellipse (0.08 and 0.5); \draw[violet!40, thick] (4, 0) ellipse (0.06 and 0.25); % Oändlighetsantydan \draw[violet!50, thick, ->] (6.5, 0.154) -- (7.2, 0.1); \draw[violet!50, thick, ->] (6.5, -0.154) -- (7.2, -0.1); \node at (7.5, 0) {$\cdots$}; % Startpunkt \node at (1, -0.3) {$1$}; % Etikett \node[violet!80!black] at (3, 1.3) {$y = \frac{1}{x}$}; % Resultat \node[draw, fill=yellow!20, rounded corners] at (4, -2.5) { \begin{tabular}{c} Volym: $V = \pi$ (ändlig!) \\ Area: $A = \infty$ (oändlig!) \end{tabular} }; \end{tikzpicture} \end{document} ``` **Beräkning:** $V = \pi \int_1^{\infty} \frac{1}{x^2} dx = \pi \left[-\frac{1}{x}\right]_1^{\infty} = \pi(0 - (-1)) = \boxed{\pi}$ --- ### 6. Sinusvågen Rotera $y = \sin(x) + 1.5$ för $0 \leq x \leq 2\pi$ kring x-axeln. ```tikz \begin{document} \begin{tikzpicture}[scale=0.85] % Axlar \draw[->, thick, gray] (-0.5,0) -- (7.5,0) node[right] {$x$}; \draw[->, thick, gray] (0,-3) -- (0,3.2) node[above] {$y$}; % Övre kontur \draw[teal!80!black, very thick, domain=0:6.28, samples=200] plot (\x, {sin(\x r) + 1.5}); % Undre kontur (spegling) \draw[teal!80!black, very thick, domain=0:6.28, samples=200] plot (\x, {-sin(\x r) - 1.5}); % Skuggning \shade[top color=teal!25, bottom color=teal!5, opacity=0.6] plot[domain=0:6.28, samples=100] (\x, {sin(\x r) + 1.5}) -- plot[domain=6.28:0, samples=100] (\x, {-sin(\x r) - 1.5}) -- cycle; % Ellipser för 3D \draw[teal!60, thick] (0, 0) ellipse (0.12 and 1.5); \draw[teal!60, thick] (1.57, 0) ellipse (0.15 and 2.5); \draw[teal!60, thick] (3.14, 0) ellipse (0.12 and 1.5); \draw[teal!60, thick] (4.71, 0) ellipse (0.08 and 0.5); \draw[teal!60, thick] (6.28, 0) ellipse (0.12 and 1.5); % Etiketter \node at (1.57, -0.3) {$\frac{\pi}{2}$}; \node at (3.14, -0.3) {$\pi$}; \node at (4.71, -0.3) {$\frac{3\pi}{2}$}; \node at (6.28, -0.3) {$2\pi$}; \node[teal!80!black] at (5, 2.5) {$y = \sin x + 1.5$}; \end{tikzpicture} \end{document} ``` --- ## Område mellan två kurvor När området mellan två funktioner roterar: ```tikz \begin{document} \begin{tikzpicture}[scale=1.3] % Axlar \draw[->, thick, gray] (-0.5,0) -- (4.5,0) node[right] {$x$}; \draw[->, thick, gray] (0,-2) -- (0,2.3) node[above] {$y$}; % Yttre kurvan f(x) - övre \draw[blue, very thick, domain=0.3:3.5, samples=100] plot (\x, {0.5*sqrt(\x)+0.5}); % Yttre kurvan f(x) - undre (spegling) \draw[blue, very thick, domain=0.3:3.5, samples=100] plot (\x, {-0.5*sqrt(\x)-0.5}); % Inre kurvan g(x) - övre \draw[red, very thick, domain=0.3:3.5, samples=100] plot (\x, {0.2*sqrt(\x)+0.2}); % Inre kurvan g(x) - undre \draw[red, very thick, domain=0.3:3.5, samples=100] plot (\x, {-0.2*sqrt(\x)-0.2}); % Skuggning av området (tvärsnittsring) \fill[green!30, opacity=0.7] (2.5, 0) ellipse (0.15 and 1.29); \fill[white] (2.5, 0) ellipse (0.15 and 0.52); \draw[green!50!black, thick] (2.5, 0) ellipse (0.15 and 1.29); \draw[green!50!black, thick] (2.5, 0) ellipse (0.15 and 0.52); % Etiketter \node[blue] at (3.8, 1.5) {$f(x)$}; \node[red] at (3.8, 0.7) {$g(x)$}; % Formel \node[draw, fill=white, rounded corners] at (2, -2.8) {$V = \pi \int_a^b \left([f(x)]^2 - [g(x)]^2\right) dx$}; \end{tikzpicture} \end{document} ``` --- ## Snabbguide — Metodval |Situation|Metod|Formel| |---|---|---| |Rotation kring **x-axeln**|Skivmetoden|$V = \pi \int_a^b [f(x)]^2 , dx$| |Rotation kring **y-axeln**|Skalmetoden|$V = 2\pi \int_a^b x \cdot f(x) , dx$| |**Mellan två kurvor**|Skivmetoden|$V = \pi \int_a^b ([f]^2 - [g]^2) , dx$| --- ## Sammanfattning ```tikz \begin{document} \begin{tikzpicture} % Skivmetoden box \node[draw, fill=blue!10, rounded corners, minimum width=6cm, minimum height=2cm] at (0, 0) { \begin{tabular}{c} \textbf{Skivmetoden} \\[0.3cm] $V = \pi \displaystyle\int_a^b [f(x)]^2 \, dx$ \end{tabular} }; % Skalmetoden box \node[draw, fill=green!10, rounded corners, minimum width=6cm, minimum height=2cm] at (8, 0) { \begin{tabular}{c} \textbf{Skalmetoden} \\[0.3cm] $V = 2\pi \displaystyle\int_a^b x \cdot f(x) \, dx$ \end{tabular} }; \end{tikzpicture} \end{document} ``` > _"Integralen summerar oändligt många infinitesimala volymelement till en exakt volym"_ --- ## Se även - [[Rotationsyta tikz]] - [[Analysens fundamentalsats]] - [[Integraler]]