Hier lernst du wie du Flächen zwischen verschiedenen Graphen oder Graphen und Koordinatenachsen bestimmst. Insbesondere das aufstellen eines passenden Integrals und die Wahl der Integrationsgrenzen sind von Bedeutung.

Flächenberechnung

Fläche zwischen zwei Funktionen

Fläche zwischen Funktion und x-Achse

Eine Fläche kann ins Unendliche reichen und dennoch einen endlichen Flächeninhalt besitzen. In diesem Fall spricht man von einem uneigentlichen Integral. Hier lernst du diese Integrale zu berechnen.

Uneigentliche Integrale

Integralgleichung

kettenregel

Lösbarkeit einer Matrix

Glechgewicht

Schiefe Ebene

Elektrische Anlage mit Abzweigunen

Flächenträgheit

Wechselstrom

Cauchy Verdichtungssatz 

Schaltvorgänge

Singulärwertzerlegung

Spann

Lineares Gleichungssystem

spannung

LR-Zerlegung mit pivotisierung 

Basis bestimmen

trigonalisierbarkeit

Fourier Koeffizienten bestimmen

Reimann-Summen

Ungleichungen 

Statik in der Ebene

Fourier-Reihen

Funktionsfolgen

Unterschied In/Homogen

Basis, Basiswechsel

Abbildungsmatrizen

Funktionen bestimmen

Geraden und Ebenen

Gruppen

lineare abbildung 

quadratische ergänzung

Baustatik 1

Anfangswertprobleme :)

Differentialgleichungen

basis 

QR-Algorithmus

QR-Zerlegung

Cholesky-Zerlegung

Bild & Urbild

Zweigstromanalyse

Transistor als Thermometer

Investment

Teilfolgen

Epsilon Delta Kriterium

prinzip der virtuellen verrückung 

Linear Kombination

Folgen, Reihen und Potenzreihen

Flächenpressung

Arbeit 

Cholesky Zerlegung

Polynomfunktionen beschreiben

Zu Maß-und Integrationstheorie: Majorisierte Konvergenz

Teilchenphysik und Festkörperphysik

Orthogonaltrajektorien

test

Taylorreihen

Regel von de L'Hopital

Mathe

KRAFT

Grenzwert Funktion 

Folgenkriterium

ϵ-δ Kriterium 

Beweis Stetigkeit 

Wahrheitstabelle

2x2 Matrix

Matrix invertieren

Basiswechselsatz

Lineare Abbildunge 

Vorderrad fahradgabel lagerkräfte berechnen 

Spannungsregelung

Jordan Normal Form

Test

Induktion 

Jordan-Normalform

Ansatz der Störfunktion zur Bestimmung der Partikulären Lösung

Kurvenintegral

dlg mithilfe inhomogenen linearen system lösen

Inverse einer Matrix

Gleichmäßige Stetigkeit

addition

Regel von L´Hospital

Lineare Abbildungen

dimension eines Untervektorraumes bestimmen

Stationäre Stoffübertragung

Polplan, Prinzip der virtuellen Verrückungen

komplexe Ungleichungen 

Bidualräume

Dualräume

Taylorpolynome

Kern und Basis

Diagonalisieren einer Matrix

Rekursive Folgen

Exponentialreihe

Darstellungsmatrizen

restglied

Basiswechsel

Gamma Funktion

Der Wert des Integrals beträgt:
\begin{align*}
\int_{0}^{\infty} \frac{d x}{\sqrt{x}(1+x)} = \pi
\end{align*}

Der Wert des Integrals beträgt:
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\int_{0}^{\infty} \frac{d x}{\sqrt{x}(1+x)} = \pi
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-10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-1228-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-1228-TEX-N-221A" d="M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z"></path><path id="MJX-1228-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-1228-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 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402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0, 76.2)"><g data-mml-node="mtd"><g data-mml-node="msubsup"><g data-mml-node="mo"><use xlink:href="#MJX-1228-TEX-LO-222B"></use></g><g data-mml-node="TeXAtom" transform="translate(1013.4, 1088.1) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use xlink:href="#MJX-1228-TEX-N-221E"></use></g></g><g data-mml-node="TeXAtom" transform="translate(556, -896.4) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use xlink:href="#MJX-1228-TEX-N-30"></use></g></g></g><g data-mml-node="mfrac" transform="translate(1937.2, 0)"><g 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Ersetze die obere Grenze des Integrals durch $a$ und bilde des Grenzwert mit $\lim _{a \rightarrow \infty}$:

Ersetze die obere Grenze des Integrals durch <mjx-container class="MathJax" jax="SVG"><svg alt="a" style="vertical-align: -0.023ex" xmlns="http://www.w3.org/2000/svg" width="1.197ex" height="1.02ex" role="img" focusable="false" viewBox="0 -441 529 451" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1222-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-1222-TEX-I-1D44E"></use></g></g></g></svg></mjx-container> und bilde des Grenzwert mit <mjx-container class="MathJax" jax="SVG"><svg alt="\lim _{a \rightarrow \infty}" style="vertical-align: -0.357ex" xmlns="http://www.w3.org/2000/svg" width="7.302ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 3227.3 851.8" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1223-TEX-N-6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z"></path><path id="MJX-1223-TEX-N-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path><path id="MJX-1223-TEX-N-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-1223-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-1223-TEX-N-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 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\begin{align*}
\int_{0}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x=\lim _{a \rightarrow \infty} \int_{0}^{a} \frac{1}{\sqrt{x}(1+x)} d x
\end{align*}

<mjx-container class="MathJax" jax="SVG"><svg alt="\begin{align*}
\int_{0}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x=\lim _{a \rightarrow \infty} \int_{0}^{a} \frac{1}{\sqrt{x}(1+x)} d x
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Nutze Substitution zum Integrieren:

\begin{align*}
&amp;&amp; z(x) &amp;=\sqrt{x} \\
\Leftrightarrow &amp;&amp; x&amp;=z^{2} \\
&amp;&amp; \frac{d z}{dx} &amp;= (\sqrt{x})^{\prime} = \frac{1}{2 \sqrt{x}} = \frac{1}{2z}\\
\Rightarrow &amp;&amp; d x &amp;= 2 z \ d z
\end{align*}

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&& z(x) &=\sqrt{x} \\
\Leftrightarrow && x&=z^{2} \\
&& \frac{d z}{dx} &= (\sqrt{x})^{\prime} = \frac{1}{2 \sqrt{x}} = \frac{1}{2z}\\
\Rightarrow && d x &= 2 z \ d z
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\begin{align*}
\lim _{a \rightarrow \infty} \int_{0}^{a} \frac{1}{\sqrt{x}(1+x)} d x &amp;=\lim _{a \rightarrow \infty} \int_{0}^{a} \frac{1}{z\left(1+z^{2}\right)} \cdot 2 z \ d z \\
&amp;=\lim _{a \rightarrow \infty} 2 \int_{0}^{a} \frac{1}{1+z^{2}} d z 
\end{align*}

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\lim _{a \rightarrow \infty} \int_{0}^{a} \frac{1}{\sqrt{x}(1+x)} d x &=\lim _{a \rightarrow \infty} \int_{0}^{a} \frac{1}{z\left(1+z^{2}\right)} \cdot 2 z \ d z \\
&=\lim _{a \rightarrow \infty} 2 \int_{0}^{a} \frac{1}{1+z^{2}} d z 
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Setze die Grenzen ein und berechne den Grenzwert:

\begin{align*}
\phantom{\lim _{a \rightarrow \infty} \int_{0}^{a} \frac{1}{\sqrt{x}(1+x)} d x }
&amp;=\lim _{a \rightarrow \infty} \left( \left.2 \arctan(z) \right|_{0}^{a} \right) \\
&amp;=\lim _{a \rightarrow \infty}(2 \arctan a-2 \arctan 0) \\
&amp;=\lim _{a \rightarrow \infty} 2 \arctan a \\
&amp;=2 \cdot \frac{\pi}{2} \\
&amp;=\pi
\end{align*}

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\phantom{\lim _{a \rightarrow \infty} \int_{0}^{a} \frac{1}{\sqrt{x}(1+x)} d x }
&=\lim _{a \rightarrow \infty} \left( \left.2 \arctan(z) \right|_{0}^{a} \right) \\
&=\lim _{a \rightarrow \infty}(2 \arctan a-2 \arctan 0) \\
&=\lim _{a \rightarrow \infty} 2 \arctan a \\
&=2 \cdot \frac{\pi}{2} \\
&=\pi
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Bestimme den Wert des folgenden uneigentlichen Integrals:
\begin{align*}
\int_{0}^{\infty} \frac{d x}{\sqrt{x}(1+x)}
\end{align*}

Bestimme den Wert des folgenden uneigentlichen Integrals:
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Dies ist eine interaktive Aufgabe zu: Uneigentliche Integrale mit praktischen Tipps zum Lösen und einer Zusammenfassung der nötigen Theorie

Anwendung der Integralrechnung - Uneigentliche Integrale | Aufgabe mit

Aufgabenstellung:

Lösungsweg:

Lösung: