Wzory:
Własności I
1)∫K * f(x)dx = k∫abf(x)dx
2)∫ab(f1x1+f1x2)dx = ∫abf1x1dx + ∫abf2x2dx
3) ∫baf(x)dx = ∫acf(x)dx + ∫cbf(x)ax
4)∫dbax = 0
5)∫aaf(x)dx = 0
6)∫abf(x)dx = −∫baf(x)dx zmiana gornej granicy na dolna
7)∫baf(x)dx = b − a
Własności II
1)∫abf(x)dx = F′(b) − F′(a) = gϵR; F′(x) = f(x)
2)∫abf(x)dx = [f(x)]ab = F′(b) − F′(a) = gϵR
Gdy f(x) ≥ o; x ∈ <a; b > to funkcja jest dodatnia → ∫abf(x)dx > 0
Gdy f(x) ≤ 0 x ∈ <a, b > to unkcja jest ujemna → ∫abf(x)dx < 0
Objętość brył obrotowych V=π∫abf2 (x)dx
Długość łuku krzywej L=$\int_{a}^{b}\sqrt{{(1 + y)}^{2}}$dx, L=$\int_{a}^{b}{f\left( x \right)dx = g > 0;y^{u} \rightarrow y^{'} \rightarrow^{v}\left( y^{'} \right)^{2} \rightarrow^{v}1 + \left( y^{'} \right)^{2} \rightarrow^{v}\text{\ \ }}\sqrt{1 + \left( y^{2} \right)} = \varphi x$
Zadania
1)$\int_{0}^{1}{x^{2}dx = \lbrack\frac{1}{3}x^{3}\rbrack_{0}^{1} = \frac{1}{3} - 0 = \frac{1}{3}}$
2) ∫163dx = 3[x]16 = 3(6−1) = 15
3) $\int_{1}^{4}{xdx = \lbrack\frac{1}{2}x^{2}\rbrack_{1}^{4} = \frac{1}{2}*16 - \frac{1}{2}*1 = \frac{15}{2}}$
4) $\int_{0}^{6}{\frac{1}{2}xdx = \frac{1}{2}\lbrack\frac{1}{2}x^{2}\rbrack_{0}^{6} = \frac{1}{4}\left( 36 - 0 \right) = 9}$
5) $\int_{2}^{4}{2xdx = 2\lbrack\frac{1}{2}x^{2}\rbrack_{2}^{4} = \lbrack x^{2}\rbrack_{2}^{4} = 16 - 4 = 12}$
6) $\int_{1}^{2}{\frac{1}{2}x^{2}dx = \frac{1}{2}\lbrack\frac{1}{3}x^{3}\rbrack_{1}^{2} = \frac{1}{6}\lbrack x^{3}\rbrack_{1}^{2} = \frac{1}{6}\left( 8 - 1 \right) = \frac{7}{6}}$
7) ∫07sinxdx = [ − cosx]07 = −[cosπ − cos0]= − (1−1) = 2
8) $\int_{2}^{3}{- x^{2}dx = - \lbrack}\frac{1}{3}x^{3}\rbrack_{2}^{3} = - \frac{1}{3}\left( 27 - 8 \right) = - \frac{19}{3}$
9) ∫0πcosxdx = [sinx]0π = (sinπ−sin0) = (0−0) = 0
S=$\int_{0}^{\pi}{|cosx|dx = \int_{0}^{\frac{\pi}{2}}{coxdx + \int_{\frac{\pi}{2}}^{\pi}{- cpsxdx = \lbrack sinx\rbrack_{\frac{\pi}{2}}^{0} - \lbrack sinx\rbrack_{\frac{\pi}{2}}^{\pi}} = \sin{\frac{\pi}{2} - sin\pi}}}$
10) $\int_{0}^{4}{\left( x - 2 \right)dx = \lbrack}\frac{1}{2}x^{2} - 2x\rbrack_{0}^{4} = \frac{1}{2}*16 - 2*4 - 0 = 8 - 8 = 0$
$S = \int_{0}^{4}{\left( x - 2 \right)dx = \int_{0}^{2}{\left( - x + 2 \right)dx + \int_{2}^{4}{\left( x - 2 \right)dx = \lbrack - \frac{1}{2}x^{2} + 2x\rbrack_{0}^{2} + \lbrack\frac{1}{2}x^{2} - 2x\rbrack_{2}^{4} = \left\lbrack - \frac{1}{2}*4 + 4 \right\rbrack + \left\lbrack \frac{1}{2}*16*8 - \frac{1}{2}*4 + 4 \right\rbrack =}}}\left( - 2 + 4 \right) + \left( - 2 + 4 \right) = 2 + 2 = 4$
11)$\ \int_{0}^{2}{\left( x^{2} - 1 \right)dx = \lbrack\frac{1}{3}x^{3}} - x\rbrack_{0}^{2} = \left\lbrack \frac{1}{3}*8 - 2 \right\rbrack = \frac{8}{3} - 2 = \frac{2}{3}$
$S = \int_{0}^{1}{\left( - x^{2} + 1 \right)dx + \int_{1}^{2}{\left( x^{2} - 1 \right)dx = \lbrack - \frac{1}{3}x^{3}}} + x\rbrack_{0}^{1} + \lbrack\frac{1}{3}x^{3} - x\rbrack_{1}^{2} = - \frac{1}{3} + 1 + \frac{1}{3}*8 - 2 - \frac{1}{3} + 1 = \frac{2}{3} + \frac{7}{3} - 1 = \frac{9}{3} - 1 = 2$
12) $\int_{1}^{3}{\frac{1}{x}dx = \lbrack ln|x|}\rbrack_{1}^{3} = \ln\left| 3 \right| - \ln\left| 1 \right| = ln3 - ln1 = ln3$
13) $\int_{0}^{2}{\left( x - 1 \right)^{2}dx = \lbrack\frac{1}{3}\left( x - 1 \right)^{3}\rbrack_{0}^{2}} = \frac{1}{3}\lbrack\left( x - 1 \right)^{3}\rbrack_{0}^{2} = \frac{1}{3}\left( 1 + 1 \right) = \frac{2}{3}$
14) $\int_{0}^{1}\frac{1}{x^{2} + 1}dx = \lbrack arctgx\rbrack_{0}^{1} = arctg1 - arctg0 = \frac{\pi}{4} - 0 = \frac{\pi}{4}$
15) S=?{ y = −x2 + 2x = x(−x+2); y = x}; −x2 + 2x = x; −x2 + 2x − x = 0; − x2 + x = 0; x(−x+1) = 0; − x + 1 = 0; x = 1|| $S = \int_{0}^{1}{\left( - x^{2} + 2x - x \right)dx = \int_{0}^{1}{\left( - x^{2} + x \right)dx = \lbrack - \frac{1}{3}x^{3}}} + \frac{1}{2}x^{2}\rbrack_{0}^{1} = - \frac{1}{3} + \frac{1}{2} = \frac{1}{6}$
16) $S = ?\left\{ y = \frac{1}{2}x + 1;y = x + 3;x = 0;x = 1 \right\}\ S = \int_{0}^{1}{\left( x + 3 - \frac{1}{2}x - 1 \right)dx = \int_{0}^{1}{\left( \frac{1}{2}x + 2 \right)dx = \lbrack\frac{1}{2} - \frac{1}{2}x^{2} + 2x}}\rbrack_{0}^{1} = \frac{1}{4} + 2 - 0 = \frac{9}{4}$
17) $S = ?\left\{ y = - x^{2} - 1;y = x - 1;x = 0;x = 2 \right\}\ S = \int_{0}^{2}( - x - 1 + x^{2} + 1)dx = \int_{0}^{2}(x^{2} - x + 2)dx = \lbrack\frac{1}{3}x^{3} - \frac{1}{2}x^{2}\rbrack_{0}^{2} = \frac{1}{3}*8 - \frac{1}{2}*4 + 4 = \frac{8}{3} + 2 = \frac{14}{3}$
18) $S = ?\left\{ y = \sqrt{x};y = 2\sqrt{x};x = 1;x = 4 \right\} S = \int_{1}^{4}(2\sqrt{x} - \sqrt{x})dx = \int_{1}^{4}\sqrt{x}dx = **\frac{2}{3}\lbrack\left( \sqrt{x} \right)^{3}\rbrack_{1}^{4}$=2/3 [$\left( \sqrt{4} \right)^{3} - \left( \sqrt{1} \right)^{3} = \frac{2}{3}\left( 8 - 1 \right) = \frac{14}{3};\ **\int_{}^{}\sqrt{x}dx = {\int_{}^{}x}^{\frac{1}{2}}dx = \frac{1}{\frac{1}{2} + 1}x^{\frac{1}{2} + 1} = \frac{\frac{1}{3}}{2}x^{\frac{3}{2}} = \frac{2}{3}\left( \sqrt{x} \right)^{3}\ $
19) $S = ?\left\{ y = x^{3};y = 2x; \right\} x^{3} = 2x;x^{3} - 2x = 0;x\left( x^{2} - 2 \right) = 0;x = 0;x1 = \sqrt{2};x2 = \sqrt{2}$; S=$\int_{0}^{\sqrt{2}}(2x - x^{3})dx = \lbrack 2*\frac{1}{2}x^{2} - \frac{1}{4}x^{4}\rbrack_{0}^{\sqrt{2}} = \lbrack x^{2} - \frac{1}{4}x^{4}\rbrack_{0}^{\sqrt{2}} = 2 - \frac{1}{4}*4 = 2 - 1 = 1$
20) S=? {y=lnx; y=x; x=1; x=e} S=$\int_{1}^{e}(x - lnx)dx = \lbrack\frac{1}{2}x^{2} - \left( xlnx - x \right)**\rbrack_{1}^{e} = \lbrack\frac{1}{2}x^{2} - xlnx + x\rbrack_{1}^{e} = \frac{1}{2}e^{2} - elne + e - \frac{1}{2} + ln1 - 1 = \frac{1}{2}e^{2} - e + e - \frac{1}{2} + 0 + 1 = \frac{1}{2}e^{2} - \frac{3}{2} = \frac{1}{2}\left( e^{2} - 3 \right) > 0;\ **\int_{}^{}{xlnxdx =}\left| f = lnx \rightarrow f^{'} = \frac{1}{x};g = 1 \rightarrow g = x \right| = xlnx - \int_{}^{}{\frac{1}{x}*xdx = x}lnx - x$
21) $S = ?\left\{ y = \frac{3}{x};y = \frac{1}{x};x = 1;x = 3 \right\} S = \int_{1}^{3}(\frac{3}{x} - \frac{1}{x})dx = \int_{1}^{3}{\frac{2}{x}dx = 2\lbrack lnx\rbrack_{1}^{3}} = 2\left( ln3 - ln1 \right) = 2ln3 = ln3^{2} = ln9$
22) S = ?{y=x2+1;y=x2+2;x=0;x=2}S = ∫02(x2 + 2 − x2 − 1)dx = ∫021dx=[x]02 = 2
23) $S = ?\left\{ y = 2x;y = 3x;y = 1;y = 2 \right\}\ D = \left\{ \ 1 \leq y + 2;\frac{1}{3}y < x \leq \frac{1}{2}y \right\}\ S = \int_{1}^{2}(\frac{1}{2}y - \frac{1}{3}y)dy = \int_{1}^{2}\frac{1}{6}ydy = \frac{1}{6}\lbrack\frac{1}{2}y^{2}\rbrack_{1}^{2} = \frac{1}{12}\left( 4 - 1 \right) = \frac{1}{12}*3 = \frac{1}{4}$
24) S = ?{y=ln|x|;y=1;y=2;x=0}D = {1≤y≤2;0≤x≤ey} S = ∫12(e3 − 0)dy = [ey]12 = e2 − e
25) $S = ?\left\{ y = 2x + 1;y = x + 3;y = 2;y = 4 \right\}\ y = x + 3 \rightarrow x = y - 3;y = 2x + 1 \rightarrow x = \frac{1}{2}y - \frac{1}{2};D = \left\{ 3 \leq y \leq 4;y - 3 \leq x \leq \frac{1}{2}y - \frac{1}{2} \right\}\ S = \ \int_{3}^{4}(\frac{1}{2}y - \frac{1}{2}) - \left( y - 3 \right)dy = \int_{3}^{4}{(\frac{1}{2}} - \frac{1}{2} - y + 3)dy = \int_{3}^{4}{( - \frac{1}{2}y} + \frac{5}{3})dy = \frac{1}{2}\lbrack - \frac{1}{2}y^{2} + \frac{5}{3}y\rbrack_{3}^{4} = \frac{1}{2}\left\lbrack - \frac{1}{2}*16 + 20 + \frac{1}{2}*9 - 15 \right\rbrack = \frac{1}{2}\left( 8 + 20 + \frac{9}{2} - 15 \right) = \frac{1}{2}\left( - 3 + \frac{9}{2} \right) = \frac{1}{2}*\frac{3}{2} = \frac{3}{4}$
26) $S = ?\left\{ y = \sqrt{x};y = 2\sqrt{x};y = 1;y = 2 \right\} S = \int_{1}^{2}{(y^{2} - \frac{y^{2}}{4})}dy = \int_{1}^{2}{(\frac{3}{4}y^{2})}dy = \frac{3}{4}\lbrack\frac{1}{3}y^{3}\rbrack_{1}^{2} = \frac{1}{4}\lbrack y^{3}\rbrack_{1}^{2} = \frac{1}{4}\left( 8 - 1 \right) = \frac{1}{4}*7 = \frac{7}{4}$
27) $S = ?\left\{ y = x;t = 3x;y = \frac{3}{x} \right\}\ D2 = \int_{0}^{\sqrt{3}}{(y - \frac{1}{3}y)}dy = \int_{0}^{\sqrt{3}}{(\frac{2}{3}y)}dy = \frac{2}{3}\int_{0}^{\sqrt{3}}(y)dy = \frac{2}{3}\lbrack\frac{1}{2}y^{2}\rbrack_{0}^{\sqrt{3}} = \frac{1}{3}\left\lbrack 3 - 0 \right\rbrack = \frac{1}{3}*3 = 1$
28) $S = ?y = x^{2};y = x^{3}\}\ x^{2} = x^{3};x^{2} - x^{3} = 0;x^{2}\left( 1 - x \right) = 0;x = 0;x = 1\ S = \int_{0}^{1}(x2 - x^{3})dx = \int_{}^{}{x^{2}dx - \int_{}^{}{x^{3}dx =}}\lbrack\frac{x^{3}}{3} - \frac{x^{4}}{4}\rbrack_{0}^{1} = \frac{1}{3} - \frac{1}{4} = \frac{1}{12}$
29) S = ?{y=x2+1;y=x2+3;x=1;x=2}S = ∫12(x2 + 3 − x2 − 1)dx = ∫122dx=2[x]12 = 2
30) $S = ?\left\{ y = x + 1;y = 2x + 3;y = 2;y = 4 \right\} S = \int_{2}^{4}(y - 1 - \frac{1}{2}y + \frac{3}{2})dy = \int_{2}^{4}{(\frac{1}{2}y} + \frac{1}{2})dy = \frac{1}{2}\int_{2}^{4}{\left( y + 1 \right)\text{dy}} = \frac{1}{2}\lbrack\frac{1}{2}y^{2} + y\rbrack_{2}^{4} = \frac{1}{2}\left( 8 + 4 - 2 - 2 \right)^{2} = 4$
31) $S = ?\left\{ y = \sqrt{x};y = \sqrt{3x};y = 1;y = 2 \right\} S = \int_{1}^{2}{\left( y^{2} - \frac{y^{2}}{3} \right)\text{dy}} = \int_{1}^{2}{\frac{2}{3}y^{2}dy = \frac{2}{3}\lbrack\frac{1}{3}y^{3}\rbrack_{1}^{2} = \frac{2}{9}\left( 8 - 1 \right) = \frac{14}{9}}$
32) $S = ?\left\{ y = lnx;y = 3lnx;y = 1;y = 3 \right\} S = \ \int_{1}^{3}(e^{y} - e^{\frac{y}{3}})dy = \int_{1}^{3}{\lbrack e^{y} - 3e^{\frac{y}{3}}\rbrack_{1}^{3} = \left( e^{3} - 3e - e + e^{\frac{1}{3}} \right) =}e^{3} - e^{\frac{1}{3}} - 4e = e^{3} - \sqrt[3]{e} - 4e$
Objętość brył obrotowych
33) V = π∫abr2dx=πr2[x]ab = πr2(b−a) = πr2 * h
34) $V = \frac{1}{3}\pi r^{2}h;V = \pi\int_{o}^{h}{\left( \frac{r}{h}x \right)^{2}dx =}\pi\frac{r^{2}}{h^{2}}\ \int_{o}^{h}{x^{2}dx =}\pi\frac{r^{2}}{h^{2}}\lbrack\frac{1}{3}x^{3}\rbrack_{o}^{h} = \frac{r^{2}}{\text{πh}^{2}}*\frac{1}{3}*h^{3} = \frac{1}{3}\pi r^{2}h$
35) $y = \frac{1}{x};x \in < 1;2 > ;V = \pi\int_{1}^{2}\frac{1}{x^{2}}dx = \pi\lbrack - \frac{1}{x}\rbrack_{1}^{2}** = \pi\left( - \frac{1}{2} + 1 \right) = \frac{\pi}{2};\ **\int_{}^{}{\frac{1}{x^{2}}\text{dx}} = \int_{}^{}{x^{2}dx =}\frac{1}{- 2 + x}x^{- 2 + 1} = \frac{1}{- 1}x^{- 1} = - \frac{1}{x}$
36) $x^{2} + y^{2} = r^{2} \rightarrow y^{2} = r^{2} - x^{2};y = \sqrt{r^{2} - x^{2}};V = 2\pi\int_{o}^{r}{\left( \sqrt{r^{2} - x^{2}} \right)^{2}dx =}2\pi\int_{0}^{r}(r^{2} - x^{2})dx = 2\pi\lbrack rx^{2} - \frac{x^{3}}{3}\rbrack_{0}^{r} = 2\pi\left( r^{3} - \frac{r^{3}}{3} \right) = 2\pi\frac{2}{3}r^{3} = \frac{4}{3}\pi r^{3}$
37) $V = ?y = \frac{1}{4}x;x \in < 1;4 > ;V = \pi\int_{1}^{4}{\frac{1}{4}x^{2}dx = \frac{1}{4}\pi\lbrack\frac{1}{3}x^{3}\rbrack_{1}^{4} = \frac{1}{12}\pi\left( 4^{3} - 1 \right) = \frac{1}{12}\pi 63 = \frac{21}{4}\pi}$
38) $V = ?y = \left( x - 2 \right)^{2}\ x \in < 1;3 > ;V = \ \pi\int_{1}^{3}{\left( x - 2 \right)^{4}dx = \pi\lbrack\frac{1}{5}\left( x - 2 \right)^{5}\rbrack_{1}^{3}** = \frac{1}{5}\pi\left( 1 + 1 \right) = \frac{2}{5}\pi;\ **\int_{}^{}{\left( x - 2 \right)^{2}dx = \frac{1}{5}\left( x - 2 \right)^{5}}}$
39) $V = ?\frac{x^{2}}{9} + y^{2} = 1\ x \in < 0;3 > ;y^{2} = 1 - \frac{x^{2}}{9};V = \pi\int_{0}^{3}{\left( 1 - \frac{x^{2}}{9} \right)dx = \pi\lbrack x - \frac{1}{9}*\frac{1}{3}x^{3}\rbrack_{0}^{3} = \pi\left( 3 - \frac{1}{27}*27 \right) = 2\pi}$
40) $v = ?\frac{x^{2}}{4} + y^{2} = 1;x \in < - 1;1 > ;y^{2} = 1 - \frac{x^{2}}{4};S = \left( 0;0 \right)a = 2,\ b = 1;V = \pi\int_{- 1}^{1}{\left( 1 - \frac{x^{2}}{4} \right)^{2}dx = 2\pi}\int_{0}^{1}(1 - \frac{x^{2}}{4})dx = 2\pi\lbrack x - \frac{1}{4}*\frac{1}{3}x^{3}\rbrack_{0}^{1} = 2\pi\left( 1 - \frac{1}{12} \right) = 2\pi*\frac{11}{12} = \frac{11\pi}{6}$
41) $V = ?y = sinx\ x \in < 0;\pi > ;V = \pi\int_{0}^{\pi}{\operatorname{}\text{xdx} =}\pi\lbrack\frac{x}{2} - \frac{1}{4}sin2x\rbrack_{0}^{\pi} = \pi\left( \frac{\pi}{2} - \frac{1}{4}sin2\pi \right)** = \frac{\pi^{2}}{2} \rightarrow \ \pi*\frac{\pi}{2};\ **\int_{}^{}{sinxdx = \frac{x}{2} - \frac{1}{4}sin2x}$
42) $V = ?\ \ y = - x - 1\ x \in < - 4;0 > ;V = \pi\int_{- 4}^{0}{\left( - x - 1 \right)^{2}dx =}\pi\int_{- 4}^{0}{\left( x + 1 \right)^{2}x = - \pi\int_{0}^{- 4}{\left( x + 1 \right)^{2}\text{dx}}} = - \pi\lbrack\frac{1}{3}\left( x + 1 \right)^{3}\rbrack_{0}^{- 4} = - \frac{\pi}{3}\lbrack\left( x + 1 \right)^{3}\rbrack_{0}^{4} = - \frac{\pi}{3}\left( - 4 + 1 \right)^{3} = - \frac{\pi}{3}\left( - 27 - 1 \right) = - \frac{\pi}{3}( - 28)$
43) $V = ?y = \sqrt{x};x = y^{2};y \in < 1;2 > ;V = \pi\int_{1}^{2}{\left( y^{2} \right)^{2}dy =}\pi\int_{1}^{2}{y^{4}dy =}\pi\lbrack\frac{1}{5}y^{5}\rbrack_{1}^{2} = \frac{\pi}{5}\left( 32 - 1 \right) = \frac{31}{5}\pi$
44) $V = ?y = \frac{1}{3}x\ y \in < 1;2 > ;x = 3y;V = \pi\int_{1}^{2}{\left( 3y \right)^{2}\text{dy}} = \pi\int_{1}^{2}{9y^{2}dy =}9\pi\lbrack\frac{1}{3}y^{3}\rbrack_{1}^{2} = 3\pi\left( 2^{3} - 1 \right) = 3\pi\left( 8 - 1 \right) = 27\pi$
45) $V = ?y = lnx;x = e^{y};y \in < 0;2 > ;V = \pi\int_{0}^{2}{\left( e^{3} \right)^{2}dy =}\pi\int_{0}^{2}{e^{2y}dy =}\pi\lbrack\frac{1}{2}e^{2y}\rbrack_{0}^{2} = \frac{\pi}{2}\left( e^{4} - e^{0} \right) = \frac{\pi}{2}(e^{4} - 1)$
46) V=? $\frac{x^{2}}{4} + y^{2} = 1;a = 2,b = 1\ xy \in < 0;1 > ;\frac{x^{2}}{4} = 1 - y^{2};x^{2} = 4\left( 1 - y^{2} \right);\ V = \pi\int_{0}^{1}{4\left( 1 - y^{2} \right)\text{dy}} = 4\pi\int_{0}^{1}{\left( 1 - y^{2} \right)\text{dx}} = 4\pi\lbrack y - \frac{1}{3}y^{3}\rbrack_{0}^{1} = 4\pi\left( 1 - \frac{1}{3}*1^{3} \right) = 4\pi\left( 1 - \frac{1}{3} \right) = 4\pi*\frac{2}{3} = \frac{8\pi}{3}$
47) $V = ?y = x^{2},\ x = \sqrt{y},\ y \in < 1;3 > ;\ \ V = \pi\int_{1}^{3}{\left( \sqrt{y} \right)^{2}\text{dy}} = \pi\int_{1}^{3}{(ydy =}\pi\lbrack\frac{1}{2}y^{2}\rbrack_{1}^{3} = \frac{\pi}{2}\left( 9 - 1 \right) = 4\pi$
48) $V = ?x^{2} - y^{2} = 4\ y \in < 0;1 > ;\frac{x^{2}}{4} - \frac{y^{2}}{4} = 1;\frac{x^{2}}{4} = 1 + \frac{y^{2}}{4};x^{2} = 4 + y^{2};S\left( 0,0 \right)a = 2,b = 0;V = \pi\int_{0}^{1}{\left( 4 + y^{2} \right)dy =}\pi\int_{0}^{1}{y^{2}dy =}4\lbrack 4*\frac{1}{3}y^{3}\rbrack_{0}^{1} = \frac{13}{3}\pi$
49) $L = ?y = 2x\ x \in < 0;1 > ;\ \ y^{'} = 2 \rightarrow \left( y^{'} \right)^{2} = 4 \rightarrow 1 + \left( y \right)^{2} = 1 + 4 = 5 \rightarrow \sqrt{1 + \left( y \right)^{2}} = \sqrt{5}$
$L = \int_{0}^{1}\sqrt{5}dx = \sqrt{5}\lbrack x\rbrack_{0}^{1} = \sqrt{5}*1 = \sqrt{5}$
50) $y = \sqrt{r^{2} - x};x \in < 0;r > ;y^{2} = r^{2} - x^{2};r^{2} = x^{2} + y^{2};L = \int_{a}^{b}\sqrt{1 + \left( y \right)^{2}}dx;y^{'} = \frac{1( - 2x)}{2\sqrt{r^{2} - x^{2}}} = \frac{- x}{\sqrt{r^{2} - x^{2}}} \rightarrow \left( y^{'} \right)^{2} = \frac{x^{2}}{r^{2} - x^{2}} \rightarrow 1 + \left( y' \right)^{2} = 1 + \frac{x^{2}}{r^{2} - x^{2}} = \ \frac{r^{2} - x^{2} + x^{2}}{r^{2} - x^{2}} = \frac{r^{2}}{r^{2} - x^{2}} \rightarrow \ \sqrt{1 + \left( y^{'} \right)^{2}} = \sqrt{\frac{r^{2}}{r^{2} - x^{2}}} = \frac{r}{\sqrt{r^{2} - x^{2}}} = \ \varphi(x)$
$L = \int_{0}^{r}{(\sqrt{\frac{r^{2}}{r^{2} - x^{2}}}})dx = r\int_{0}^{r}\frac{1}{\sqrt{r^{2} - x^{2}}}dx = r\lbrack\arcsin\frac{x}{r}\rbrack_{0}^{r} = r\left( arcsin1 - arcsin0 \right) = r\left( \frac{\pi}{2} - 0 \right) = \frac{\text{πr}}{2}\ $
$L\ okregu = 4*\frac{\pi r}{2} = 2\pi r$
51) $y = \frac{1}{3}x + 2\ x \in < - 3;6 > ;\left( y^{'} \right) = - \frac{1}{3}\left( y^{'} \right)^{2} = \frac{1}{9} \rightarrow 1 + \left( y^{'} \right)^{2} = 1\frac{1}{9} = \frac{10}{9} \rightarrow \sqrt{1 + \left( y^{'} \right)^{2}} = \sqrt{\frac{10}{9}} = \frac{\sqrt{10}}{3}$
$L = \int_{- 3}^{6}{\frac{\sqrt{10}}{3}dx =}\frac{\sqrt{10}}{3}\lbrack x\rbrack_{- 3}^{6} = \frac{\sqrt{10}}{3}\left( 6 + 3 \right) = \frac{\sqrt{10}}{3}*9 = 3\sqrt{10}$
52) $y = 2x - \frac{1}{2};x \in < - 2;4 > ;2x - \frac{1}{2} = 0;x = \frac{1}{4};y^{'} = 2 \rightarrow \left( y^{'} \right)^{2} - 4 \rightarrow 1 + \left( y^{'} \right)^{2} = 5 \rightarrow \sqrt{1 + \left( y^{'} \right)^{2}} = \sqrt{5};$
$L = \int_{- 2}^{4}{\sqrt{5}\text{dx}} = \sqrt{5}\lbrack x\rbrack_{- 2}^{4} = \sqrt{5}\left( - 4 + 2 \right) = 6\sqrt{5}$
53) $y = \sqrt{x^{3}};x \in < 0;1 > ;y = x^{\frac{3}{2}} = \left( \sqrt{x} \right)^{3} = \sqrt{x}*\sqrt{x}*\sqrt{x} = x\sqrt{x};y^{'} = \frac{1}{2\sqrt{x^{3}}}*3x^{2};y^{'} = \left( x^{\frac{3}{2}} \right)^{'} = \frac{3}{2}x^{\frac{1}{2}} \rightarrow \left( y^{'} \right)^{2} = \left( \frac{3}{2}*x^{\frac{1}{2}} \right)^{2} = \frac{9}{4}*x = \ \sqrt{4x - x^{2}} = 1 + \frac{9}{4}x \rightarrow \sqrt{1 + \left( y^{'} \right)^{2}} = \sqrt{1 + \frac{9}{4}x}$
$L = \int_{0}^{1}\sqrt{1 + \frac{9}{4}x}dx = \lbrack\frac{8}{27}\left( \sqrt{\frac{13}{4}} \right)^{3}\rbrack_{0}^{1}** = \frac{8}{27}\left\lbrack \left( \sqrt{\frac{13}{4}} \right)^{3} - 1 \right\rbrack;**\int_{}^{}{\sqrt{1 + \frac{9}{4}x}\text{dx}} = \left| 1 + \frac{9}{4}x = t^{2};\frac{9}{4}dx = 2tdt;dx = \frac{8}{9}\text{tdt} \right| = \int_{}^{}{t*\frac{8}{9}tdt = \frac{8}{9}\int_{}^{}{t^{2}dt = \frac{8}{9}*\frac{1}{3}t^{3} = \frac{8}{27}t^{3} = \frac{8}{26}\left( \sqrt{\frac{13}{4}} \right)^{3}}}$
54) $y = \sqrt{4x - x^{2}};x \in < 0;2 > ;S = \left( 2,0 \right)r = 2;\ y = \sqrt{4x - x^{2}};y^{2} = {\sqrt{4x - x^{2}}}^{2};y^{2} + x^{2} - 4x = 0;y^{2} + \left( x - 2 \right)^{2} - 4 = 0;\left( x - 2 \right)^{2} + \left( y - 0 \right)^{2} = 2^{2};y^{'} = \left( \sqrt{4x - x^{2}} \right)^{'} = \frac{1}{2\sqrt{4x - x^{2}}}*\left( 4 - 2x \right) = \frac{8(2 - x)}{2\sqrt{4x - x^{2}}} = \frac{2 - x}{\sqrt{4x - x^{2}}};$
$\left( y^{'} \right)^{2} = \frac{2 - x}{\sqrt{4x - x^{2}}} \rightarrow 1 + \left( y^{'} \right)^{2}\ \frac{\left( 2 - x \right)^{2}}{4x - x^{2}} + 1 = \frac{4 - 4x + x^{2} + 4x - x^{2}}{4x - x^{2}} = \frac{4}{4x - x^{2}} \rightarrow \sqrt{\frac{4}{4x - x^{2}}} = \frac{2}{\sqrt{4x - x^{2}}}\ $
$L = \int_{0}^{2}\frac{2}{\sqrt{4x - x^{2}}}dx = 2\lbrack arcsin\frac{x - 2}{2}\rbrack_{0}^{2}** = 2\left( arcsin0 - arcsin2 \right) = 2\left\lbrack o - \left( - \frac{\pi}{2} \right) \right\rbrack = 2*\frac{\pi}{2} = \pi;\ **\int_{}^{}\frac{1}{\sqrt{4x - x^{2}\text{\ \ }}}dx = \int_{}^{}\frac{1}{\sqrt{4 - \left( x - 2 \right)^{2}}}dx = arcsin\frac{x - 2}{2}$
55) $L = ?y = \frac{1}{2}x - \frac{1}{4}\ x \in < - 2;2 > ;y^{'} = \frac{1}{2} \rightarrow \left( y^{'} \right)^{2} = \frac{1}{4} \rightarrow 1 + \left( y^{2} \right)^{'} = 1 + \frac{1}{4} = \frac{5}{4} \rightarrow \frac{\sqrt{5}}{2}$
$L = \int_{- 2}^{2}\frac{\sqrt{5}}{2}dx = \frac{\sqrt{5}}{2}\lbrack x\rbrack_{- 2}^{2} = \frac{\sqrt{5}}{2}\left( 2 + 2 \right) = \frac{\sqrt{5}}{2} - 4 = 2\sqrt{5};y = \frac{1}{2}x - \frac{1}{2};y + \frac{1}{2} = \frac{1}{2}x;2y + 1 = x;x = 2y + 1$
L= $\int_{\frac{- 3}{2}}^{\frac{1}{2}}\sqrt{5}dy = \sqrt{5}\lbrack y\rbrack_{- \frac{3}{2}}^{\frac{1}{2}} = \sqrt{5}\left( \frac{1}{2} + \frac{3}{2} \right) = 2\sqrt{5};\ \ x^{'} = 2 \rightarrow \left( x^{'} \right)^{2} = 4 \rightarrow \left( x^{'} \right)^{2} + 1 = 5 \rightarrow \sqrt{5}$
Pole z obszarami ogranoczonymi
56) $S = ?\left\{ y = x + 1;y = x^{2} + 2;x = 0;x = 1 \right\}\ S = \int_{0}^{1}{(x^{2} + 2 - x + 1)}dx = \int_{0}^{1}{\left( x^{2} - x + 3 \right)dx =}\lbrack\frac{1}{3}x^{3} - \frac{1}{2}x^{2} + 3x\rbrack_{0}^{1} = \frac{1}{3} - \frac{1}{2} + 3 = \frac{2 - 3 + 18}{6} = \frac{17}{6}$
57) $S = ?\left\{ y = \sqrt{x};y = 3\sqrt{x};x = 4 \right\}\ S = \int_{0}^{4}{(3\sqrt{x} -}\sqrt{x})dx = \int_{0}^{4}{2\sqrt{x}dx =}2\int_{0}^{4}{\sqrt{x}dx =}2\int_{0}^{4}{x^{\frac{1}{2}}dx = \ }2\lbrack\frac{1}{\frac{1}{2} + 1}x^{\frac{1}{2} + 1}\rbrack_{0}^{4} = 2\lbrack\frac{2}{3}x^{\frac{3}{2}}\rbrack_{0}^{4} = \frac{4}{3}\lbrack x^{\frac{3}{2}}\rbrack_{0}^{4} = \frac{4}{3}\lbrack\left( \sqrt{x} \right)^{3}\rbrack_{0}^{4} = \frac{4}{3}\left( 2^{3} - 0 \right) = \frac{32}{3}$
58) S = ?{y=x+1;y=x+2;y=3;y=5}S = ∫35(y−1−y+2)dy = ∫351dy = [y]35 = [5−3] = 2
59)$S = ?\left\{ y = lnx;y = 2x + 1 \rightarrow x = \frac{1}{2}y - \frac{1}{2};y = 1;y = 2 \right\} S = \ \int_{1}^{0}{\left( e^{y} - \frac{1}{2}y + \frac{1}{2} \right)\text{dy}} = \lbrack e^{3} - \frac{1}{2}*\frac{1}{2}y^{2}\rbrack_{1}^{2} = \lbrack e^{y} - \frac{1}{4}y^{2} + \frac{1}{2}y\rbrack_{1}^{2} = e^{2} - 1 + 1 - e + \frac{1}{4} = e^{2} - e - \frac{1}{4} > 0$
60) $S = ?\left\{ y = x^{2} + 1;y = x + 1 \right\} x^{2} + 1 = x + 1;x^{2} = x;x\left( x - 1 \right) = 0;x = 0;x = 1;S = \int_{0}^{1}{\left( x + 1 - x^{2} - 1 \right)dx =}\int_{0}^{1}{\left( - x^{2} + x \right)\text{dx}} = \lbrack - \frac{1}{3}x^{3} + \frac{1}{2}x^{2}\rbrack_{0}^{1} = - \frac{1}{3} + \frac{1}{2} = \frac{1}{6}$
61) $S = ?\left\{ y = x^{2} + x + 2;y = - x^{2} + 2x + 2 \right\} x^{2} + x + 2 = - x^{2} + 2x + 2;2x^{2} - x = 0;x\left( 2x - 1 \right) = 0;x = 0;x = \frac{1}{2};\ \ S = \int_{0}^{\frac{1}{2}}{( - x^{2} + x + 2 - x^{2} - x - 2)}dx = \int_{0}^{\frac{1}{2}}{\left( - 2x^{2} + x \right)dx =}\lbrack - 2*\frac{1}{3}x^{3} + \frac{1}{2}x^{2}\rbrack_{0}^{\frac{1}{2}} = - \frac{2}{3}*\frac{1}{8} + \frac{1}{2}*\frac{1}{4} = \frac{1}{8} + \frac{1}{3} = \frac{1}{24}$