Geometrical Application of Definite Integral and Application of Riemann Integral in Mechanics and Physics.

1. The area of a plane figure. The area of a figure bounded by the graph of a continuous function $y=f(x),,(f(x)\ge 0),$ two straight lines $x=a$ and $x=b$ and the $x$-axis, or the area of a curvilinear trapezoid bounded by the arc of the graph of the function $y=f(x),,,a\le x\le b$ (see Fig. 1) is calculated by the formula $$S=\underset{a}{\overset{b}{\int }}f(x)dx.$$

The area of a figure bounded by the graphs of continuous functions $y=f_1(x)$ and $y=f_2(x),,,f_1(x)\le f_2(x)$ and two straight lines $x=a$, $x=b$ (see Fig. 2) is determined by the formula $S=\underset{a}{\overset{b}{\int }}(f_2(x)-f_1(x)),dx$.

If the figure is bounded by a curve with parametric equations $x=x(t),,y=y(t),$ straight lines $x=a,,,x=b$ and the $x$-axis, then its area is calculated by the formula$$S=\underset{t_1}{\overset{t_2}{\int }}y(t)x^{\prime }(t)dt=\underset{t_1}{\overset{t_2}{\int }}y(t)dx(t),(1)$$where the limits of integration are found from the equations $a=x(t_1),,,b=x(t_2),,$($y(t)\ge 0$ on the interval $[t_1,,t_2]$).

Formula (1) is also applicable for calculating the area of a figure bounded by a closed curve (the parameter $t$ should vary from $t_1$ to $t_2$ in a counterclockwise direction).

The area of a figure bounded by the graph of a continuous function $r=r(\phi )$ and two rays $\phi =\alpha ,$ $\phi =\beta ,$ where $\phi$ and $r$ are polar coordinates, or the area of a curvilinear sector bounded by the arc of the function graph, $r=r(\phi ),,,\alpha \le \phi \le \beta ,$ is calculated by the formula $$S=\frac{1}{2}\underset{\alpha }{\overset{\beta }{\int }}{r}^{2}d\phi .$$

2. The length of a curve's arc.

If a smooth curve is given by the equation $y=f(x),$ then the length $l$ of its arc is given by the formula $$l=\underset{a}{\overset{b}{\int }}\sqrt{1+(y^{\prime }{)}^2}dx,$$

Where $a$ and $b$ are the abscissas of the ends of the arc.

If the curve is defined by parametric equations $x=x(t),,y=y(t)(t_1\le t\le t_2),$ then

$$l=\underset{t_1}{\overset{t_2}{\int }}\sqrt{(x_t^{\prime }{)}^2+(y_t^{\prime }{)}^2}dt.$$

Similarly, the length of a curve in space defined by parametric equations $x=x(t),,y=y(t),,,z=z(t),,,t_1\le t\le t_2:$ $$l=\underset{t_1}{\overset{t_2}{\int }}\sqrt{(x_t^{\prime }{)}^2+(y_t^{\prime }{)}^2+(z_t^{\prime }{)}^2}dt.$$

If a smooth curve is defined by a polar equation $r=r(\phi ),$ $\alpha \le \phi \le \beta ,$ then$$l=\underset{\alpha }{\overset{\beta }{\int }}\sqrt{r^2+(r^{\prime }{)}^2}d\phi .$$

3. Surface area of revolution.

The surface area formed by revolving the arc of a curve, defined by the function $y=f(x),,,a\le x\le b,$ around the $x$-axis is calculated by the formula $$Q_x=2\pi \underset{a}{\overset{b}{\int }}f(x)\sqrt{1-(f^{\prime }(x){)}^2}dx,$$

If the arc is defined by parametric equations $x=x(t),,y=y(t)(t_1\le t\le t_2),$ then

$$Q_x=2\pi \underset{t_1}{\overset{t_2}{\int }}y(t)\sqrt{(x_t^{\prime }{)}^2+(y_t^{\prime }{)}^2}dt.$$

If the arc is given in polar coordinates as $r=r(\phi ),$ $\alpha \le \phi \le \beta ,$ then$$Q_x=2\pi \underset{\alpha }{\overset{\beta }{\int }}r\sin \phi \sqrt{r^2+(r^{\prime }{)}^2}d\phi .$$

If the arc of the curve rotates around an arbitrary axis, then the surface area of revolution is expressed by an integral. $$Q=2\pi \underset{A}{\overset{B}{\int }}Rdl,$$ where $R-$ The distance from a point on the curve to the axis of rotation, $dl$ being the infinitesimal arc length, $A$ and $B$ being the limits of integration corresponding to the ends of the arc. In this case, $R$ and $dl$ should be expressed in terms of the integration variable.

4. Volume of a solid.

If the area $S(x)$ of the cross-section of a solid perpendicular to the $x$-axis is a continuous function on the interval $[a,b]$, then the volume of the solid is calculated by the formula

$$V=\underset{a}{\overset{b}{\int }}S(x)dx.$$

The expression for the function $S(x)$ is quite straightforward in the case of bodies of revolution. For example, if a curvilinear trapezoid bounded by the curve $y=f(x),,,a\le x\le b$ rotates around the $x$-axis or the $y$-axis, then the volumes of the bodies of revolution are calculated, respectively, by the formulas:

$$V_x=\pi \underset{a}{\overset{b}{\int }}{f}^2(x)dx,$$

$$V_y=2\pi \underset{a}{\overset{b}{\int }}x|f(x)|dx,a\ge 0.$$

If a curvilinear sector bounded by the curve $r=r(\phi )$ and the rays $\phi =\alpha$ and $\phi =\beta$ rotates around the polar axis, then the volume of the solid of revolution is equal to $$V=\frac{2}{3}\pi {\int }_{\alpha }^{\beta }{r}^3\sin \phi d\phi .$$

5. Moments and centers of mass of plane curves.

If the arc of a curve is given by the equation $y=f(x),a\le x\le b,$ and has a density $\rho =\rho (x),$ then the statistical moments of this arc $M_x$ and $M_y$ with respect to the coordinate axes $Ox$ and $Oy$ are equal to $$M_x=\underset{a}{\overset{b}{\int }}\rho (x)f(x)\sqrt{1+(f^{\prime }(x){)}^2}dx,$$ $$M_y=\underset{a}{\overset{b}{\int }}\rho (x)x\sqrt{1+(f^{\prime }(x){)}^2}dx,$$

The moments of inertia $I_x$ and $I_y$ with respect to the same axes $Ox$ and $Oy$ are calculated by the formulas$$I_x=\underset{a}{\overset{b}{\int }}\rho (x)f^2(x)\sqrt{1+(f^{\prime }(x){)}^2}dx,$$ $$I_y=\underset{a}{\overset{b}{\int }}\rho (x)x^2\sqrt{1+(f^{\prime }(x){)}^2}dx,$$

and the coordinates of the center of mass $x$ and $y$ by the formulas $$\tilde{x}=\frac{M_y}{l}=\frac{1}{l}\underset{a}{\overset{b}{\int }}\rho (x)x\sqrt{1+(f^{\prime }(x){)}^2}dx,$$

$$\tilde{y}=\frac{M_x}{l}=\frac{1}{l}\underset{a}{\overset{b}{\int }}\rho (x)f(x)\sqrt{1+(f^{\prime }(x){)}^2}dx,$$

where the mass of the arc, i.e.,

$$I=\underset{a}{\overset{b}{\int }}\rho (x)\sqrt{1+(f^{\prime }(x){)}^2}dx,$$

6. Physical problems.

The distance traveled by an object with velocity $v(t)$ over the time interval $[t_1,t_2]$ is expressed by the integral$$S=\underset{t_1}{\overset{t_2}{\int }}v(t)dt.$$

The work done by a variable force $f(x)$ acting along the $Ox$ axis over the interval $[a,b]$ is expressed by the integral $$A=\underset{a}{\overset{b}{\int }}f(x)dx.$$

Tags: Geometrical Application of Definite Integral, Riemann Integral, Surface area of revolution, The length of a curve's arc, calculus, integral, mathematical analysis