07 Integral­rechnung

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1. Riemann Sums and the Definite Integral

Integral calculus fundamentally deals with accumulating quantities, often visualized as finding the area under a curve.

1.1. Motivation and Historical Context

• Ancient Roots: Archimedes (around 250 BCE) calculated the area of a parabolic segment, effectively performing an early integration.

• 17th Century Development: Newton and Leibniz independently developed calculus, establishing the crucial link between differentiation (tangent problem) and integration (area problem).

• 19th Century Rigor: Cauchy and Weierstrass provided the formal definitions and proofs used today, including defining the derivative as a limit of the difference quotient and rigorously proving the Fundamental Theorem 3333. Bernhard Riemann formalized the concept of the integral using sums44.

1.2. The Riemann Sum

Many practical problems lead to approximating a total quantity by summing up small pieces:

• Distance from Velocity: Total distance is approximated by summing distances traveled in small time intervals: $s\approx \sum v(t_i)\Delta t$5.

• Volume: The volume of a solid (like a sphere) is approximated by summing volumes of thin slices (like cylinders): $V_{Kugel}\approx \sum \pi (r^2-x_i^2)\Delta x$6.

• Work: The work done (e.g., by a pump) is approximated by summing the work done on small portions: $A\approx \sum (\pi r^2\rho g(x_i+h)\Delta x)$7.

These sums all take the general form ${\sum }_{i=0}^{n-1}f(x_i)\Delta x$, known as a Riemann Sum888. The exact value is obtained by taking the limit as the width of the pieces goes to zero ($\Delta x\to 0$ or $n\to \infty$)9.

1.3. The Definite Integral

• Definition: The definite integral of a function $f(x)$ from $a$ to $b$ is defined as the limit of the Riemann Sum:

  ${\int }_a^{b}f(x)dx:={\lim }_{\Delta x\to 0\text{ bzw. }n\to \infty }\left({\sum }_{i=0}^{n-1}f(x_i)\Delta x\right)$

• Terminology:

  • $a,b$: Integration limits (lower and upper).

  • $\int$: Integral sign (originating from Leibniz’s notation 12).

  • $f(x)$: Integrand.

  • $dx$: Indicates the integration variable.

• Result: A definite integral is always a number.

1.4. Geometric Interpretation: Area

• If $f(x)\ge 0$ on $[a,b]$, the Riemann sum approximates the area under the curve using rectangles (Untersumme/Obersumme).

• In this case, the definite integral ${\int }_a^{b}f(x)dx$ gives the exact area under the graph of $f(x)$ from $x=a$ to $x=b$.

• Important: If $f(x)$ takes negative values, the integral represents a net area, where areas below the x-axis are counted negatively.

  • If $f(x)<0$ on $[a,b]$, then ${\int }_a^{b}f(x)dx=-I$ (negative of the geometric area).

1.5. Direct Calculation (via Definition)

Calculating integrals directly from the Riemann sum definition is possible but often tedious, involving sums and limits 18.

• Example: ${\int }_0^{5}xdx$

  • Geometrically, this is the area of a triangle: $\frac{1}{2}\cdot 5\cdot 5=12.5$ 19.

  • Via definition: $\Delta x=5/n$, $x_i=i\cdot (5/n)$20.

  • ${\int }_0^{5}xdx={\lim }_{n\to \infty }{\sum }_{i=0}^{n-1}(i\frac{5}{n})\frac{5}{n}={\lim }_{n\to \infty }(\frac{5}{n}{)}^2{\sum }_{i=0}^{n-1}i$ 21.

  • Using ${\sum }_{i=0}^{n-1}i=\frac{(n-1)n}{2}$ gives ${\lim }_{n\to \infty }\frac{25}{n^2}\frac{n(n-1)}{2}={\lim }_{n\to \infty }12.5\frac{n^2-n}{n^2}=12.5$ 22.

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2. Properties and the Fundamental Theorem of Calculus

2.1. Basic Definitions and Properties

• Definitions 23:

  • ${\int }_a^{a}f(x)dx:=0$

  • ${\int }_b^{a}f(x)dx:=-{\int }_a^{b}f(x)dx$

• Linearity 24:

  • ${\int }_a^{b}k\cdot f(x)dx=k\cdot {\int }_a^{b}f(x)dx$ (Factor rule)

  • ${\int }_a^b(f(x)\pm g(x))dx={\int }_a^{b}f(x)dx\pm {\int }_a^{b}g(x)dx$ (Sum rule)

• Interval Additivity 25: For $a\le b\le c$:

  • ${\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx={\int }_a^{c}f(x)dx$

• Mean Value Theorem for Integrals 26: If $f$ is continuous on $[a,b]$, there exists a $c\in [a,b]$ such that:

  • ${\int }_a^{b}f(x)dx=f(c)\cdot (b-a)$.

  • Geometrically: The area under the curve equals the area of a rectangle with width $(b-a)$ and some height $f(c)$ attained within the interval 27.

2.2. Antiderivatives (Stammfunktionen)

• Definition: A function $F(x)$ is called an antiderivative (Stammfunktion) of $f(x)$ if $F^{\prime }(x)=f(x)$28.

• If $F(x)$ is one antiderivative, then any other antiderivative must be of the form $F(x)+c$ for some constant $c$29.

• Indefinite Integral: The indefinite integral $\int f(x)dx$ represents the set of all antiderivatives of $f(x)$ 30.

  • Example: $\int x^{3}dx=\frac{1}{4}{x}^4+c$ 31.

2.3. The Fundamental Theorem of Calculus (Hauptsatz)

This theorem provides the crucial link between differentiation and integration, enabling the efficient calculation of definite integrals32323232.

Part 1 (1. Fassung)

• Defines the integral function $F(\xi )$ with a variable upper limit33:

  $F(\xi ):={\int }_0^{\xi }f(x)dx$

  (Lower limit could be any constant $a$).

• Statement: The derivative of this integral function with respect to its upper limit is the original function evaluated at that limit 34:

  $\frac{dF}{d\xi }=\frac{d}{d\xi }\left({\int }_0^{\xi }f(x)dx\right)=f(\xi )$

• Interpretation: Integration (as building the integral function) is the inverse operation of differentiation35. The rate of change of the accumulated area up to $\xi$ is precisely the value of the function at $\xi$ 36.

Part 2 (2. Fassung) - The Evaluation Theorem

• Statement: If $F(x)$ is any antiderivative of $f(x)$ (i.e., $F^{\prime }(x)=f(x)$), then the definite integral can be calculated by evaluating $F$ at the limits 37:

  ${\int }_a^{b}f(x)dx=F(b)-F(a)$

• Notation: $F(b)-F(a)$ is often abbreviated as $F(x){|}_a^b$ 38.

• Algorithm for Calculation 39393939:

  1. Find an antiderivative $F(x)$ for the integrand $f(x)$.

  2. Calculate $F(b)-F(a)$.

• Example: ${\int }_1^7{x}^{2}dx$

  1. An antiderivative of $x^2$ is $F(x)=\frac{1}{3}{x}^3$40.

  2. ${\int }_1^7{x}^{2}dx=F(7)-F(1)=\frac{1}{3}(7^3)-\frac{1}{3}(1^3)=\frac{343}{3}-\frac{1}{3}=\frac{342}{3}=114$ 41.

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3. Integration Rules and Techniques

Finding antiderivatives is the key challenge. Basic rules come from reversing differentiation rules. More complex functions require specific techniques.

3.1. Basic Integration Rules (Reversing Differentiation)

Based on known derivatives 42:

• Linearity: $\int (f\pm g)dx=\int fdx\pm \int gdx$ and $\int kfdx=k\int fdx$ 43.

• Power Rule: $\int x^{k}dx=\frac{1}{k+1}{x}^{k+1}+c$ (for $k\ne -1$) 44.

• Special Case $k=-1$: $\int \frac{1}{x}dx=\ln |x|+[cit{e}_{s}tart]c$ (Note: Original text states $x>0$ for $\ln (x)$, but $\ln |x|$ is more general) 45.

• Trigonometric: $\int \sin (x)dx=-\cos (x)+c$ and $\int \cos (x)dx=\sin (x)+c$ (for $x$ in radians) 46.

• Exponential: $\int e^{x}dx=e^x+c$ and $\int a^{x}dx=\frac{1}{\ln (a)}{a}^x+c$ (for $a>0,a\ne 1$) 47.

3.2. Integration by Parts (Partielle Integration)

This technique comes from reversing the product rule for differentiation48484848.

• Product Rule: $(u\cdot v{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$49.

• Integrating both sides gives $u\cdot v=\int u^{\prime }vdx+\int u{v}^{\prime }dx$ 50.

• Rule (Indefinite):

  $\int u^{\prime }(x)v(x)dx=u(x)v(x)-\int u(x)v^{\prime }(x)dx+c$

  51

• Rule (Definite):

  ${\int }_a^b{u}^{\prime }(x)v(x)dx=[u(x)v(x){]}_a^b-{\int }_a^{b}u(x)v^{\prime }(x)dx$

  52

• Strategy: The goal is to choose $u^{\prime }$ and $v$ such that the new integral $\int u{v}^{\prime }dx$ is easier to solve than the original $\int u^{\prime }vdx$ 53.

• Example: $\int x\cos (x)dx$

  • Choice 1: $u^{\prime }=x,v=\cos (x)⟹u=\frac{1}{2}{x}^2,v^{\prime }=-\sin (x)$. Integral becomes $\int \frac{1}{2}{x}^2(-\sin x)dx$, which is harder 54.

  • Choice 2: $u^{\prime }=\cos (x),v=x⟹u=\sin (x),v^{\prime }=1$. Rule gives:

    • $\int x\cos (x)dx=\sin (x)\cdot x-\int \sin (x)\cdot 1dx$ 55.

    • $=x\sin (x)-(-\cos (x))+c=x\sin (x)+\cos (x)+c$56.

3.3. Integration by Substitution

This technique comes from reversing the chain rule and is useful for integrals involving composite functions57. It often applies when the integrand contains a function $g(x)$ and its derivative $g^{\prime }(x)$ as a factor 58585858.

• Chain Rule: $(F(g(x)){)}^{\prime }=F^{\prime }(g(x))\cdot g^{\prime }(x)$59. Let $f=F^{\prime }$.

• Integrating gives $\int f(g(x))g^{\prime }(x)dx=F(g(x))+c$. Since $F$ is an antiderivative of $f$, $F(y)=\int f(y)dy$.

• Rule (Indefinite):

  $\int f(g(x))g^{\prime }(x)dx=\int f(y)dy{|}_{y=g(x)}$

  60

• Rule (Definite): The integration limits must also be transformed:

  ${\int }_a^{b}f(g(x))g^{\prime }(x)dx={\int }_{g(a)}^{g(b)}f(y)dy$

  61

• Practical Use (“u-Substitution”):

  1. Identify an inner function $u=g(x)$ such that its derivative $g^{\prime }(x)$ (or a constant multiple) is also present.

  2. Find $du=g^{\prime }(x)dx$.

  3. Substitute $u$ and $du$ into the integral, converting it entirely in terms of $u$.

  4. Change integration limits to $g(a)$ and $g(b)$ (for definite integrals).

  5. Integrate with respect to $u$.

  6. Substitute back $u=g(x)$ (for indefinite integrals).

• Example: $\int \tan (x)dx=\int \frac{\sin (x)}{\cos (x)}dx$

  1. Let $u=\cos (x)$62626262.

  2. $du=-\sin (x)dx$63.

  3. Substitute: $\int \frac{-du}{u}=-\int \frac{1}{u}du$ 64.

  4. Integrate: $-\ln |u|+[cit{e}_{s}tart]c$ 65.

  5. Substitute back: $-\ln |\cos (x)|+c=\ln |\sec (x)|+[cit{e}_{s}tart]c$ 66.

3.4. Non-Elementarily Integrable Functions

• Not every elementary function has an antiderivative that can be expressed using elementary functions (polynomials, roots, trig, exp, log)67676767. Such functions are not closed (or elementarily) integrable 68.

• Examples: $e^{-x^2}$, $\frac{\sin (x)}{x}$, $\frac{e^x}{x}$, $\sqrt{1+x^4}$ 69.

• Risch Algorithm: A complex (semi-)algorithm exists to determine if a function has an elementary antiderivative and find it if it does 70. It’s implemented in computer algebra systems71.

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4. Applications: Lengths, Areas, and Volumes

Definite integrals provide a powerful tool for geometric calculations.

4.1. Area Calculation

• Area between graph and x-axis: If $f(x)\ge 0$ on $[a,b]$, Area = ${\int }_a^{b}f(x)dx$. If $f(x)\le 0$, Area = $-{\int }_a^{b}f(x)dx=|{\int }_a^{b}f(x)dx|$ 72. If $f(x)$ changes sign, integrate positive and negative parts separately and add the absolute values.

• Area between two curves: If $f(x)\ge g(x)$ on $[a,b]$, the area between the curves is:

  $A={\int }_a^b(f(x)-g(x))dx$

  • First find intersection points $a,b$ by solving $f(x)=g(x)$ 73.

  • Example: Area between $f(x)=-x^2+5$ and $g(x)=0.5x+2$. Intersections at $x=-2,x=1.5$. Area = ${\int }_{-2}^{1.5}((-x^2+5)-(0.5x+2))dx={\int }_{-2}^{1.5}(-x^2-0.5x+3)dx=\frac{343}{48}$74.

4.2. Arc Length

• The length $\Lambda$ of a curve $y=f(x)$ from $x=a$ to $x=b$ is found by integrating the length element $ds=\sqrt{d{x}^2+d{y}^2}=\sqrt{1+(f^{\prime }(x){)}^2}dx$.

• Formula:

  $\Lambda ={\int }_a^b\sqrt{1+(f^{\prime }(x){)}^2}dx$

  75(Derived by approximating the curve with small line segments and taking the limit 76).

4.3. Volume of Revolution

• If a region bounded by $y=f(x)$, the x-axis, $x=a$, and $x=b$ is rotated around the x-axis, it generates a solid of revolution77.

• The volume is calculated using the disk method: Sum the volumes of infinitesimally thin disks with radius $f(x_i)$and thickness $\Delta x$. Volume of disk $\approx \pi [f(x_i){]}^2\Delta x$ 78.

• Formula (Rotation around x-axis):

  $V=\pi {\int }_a^b[f(x){]}^{2}dx$

  79

• Examples:

  • Cone: Generated by rotating $f(x)=\frac{r}{h}x$ from $0$ to $h$. $V=\pi {\int }_0^h(\frac{r}{h}x{)}^{2}dx=\pi \frac{r^2}{h^2}[\frac{1}{3}{x}^3{]}_0^h=\frac{1}{3}\pi r^{2}h$ 80.

  • Sphere: Generated by rotating $f(x)=\sqrt{r^2-x^2}$ from $-r$ to $r$. $V=\pi {\int }_{-r}^r(r^2-x^2)dx=\pi [r^{2}x-\frac{1}{3}{x}^3{]}_{-r}^r=\frac{4}{3}\pi r^3$ 81.

  • Gabriel’s Horn: Rotate $y=1/x$ from $x=1$ to $\infty$. This uses an improper integral:

    • $V={\lim }_{b\to \infty }\pi {\int }_1^b(\frac{1}{x}{)}^{2}dx={\lim }_{b\to \infty }\pi [-\frac{1}{x}{]}_1^b={\lim }_{b\to \infty }\pi (-\frac{1}{b}-(-\frac{1}{1}))=\pi (0+1)=\pi$ 82828282.

    • Paradoxically, this infinite horn has a finite volume ($\pi$) but an infinite surface area 83.