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Resolución de integrales/ 2-y^2/ √(r^2-y^2)

Integral de √(r^2-y^2) dy

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |     _________   
 |    /  2    2    
 |  \/  r  - y   dy
 |                 
/                  
0
∫01r2−y2 dy\int\limits_{0}^{1} \sqrt{r^{2} - y^{2}}\, dy 
Integral(sqrt(r^2 - y^2), (y, 0, 1))
Respuesta (Indefinida) [src] 
                         //     2      /y\                                                        \
                         ||  I*r *acosh|-|              3                                 | 2|    |
                         ||            \r/           I*y                 I*r*y            |y |    |
                         ||- ------------- + ------------------- - -----------------  for |--| > 1|
                         ||        2                   _________           _________      | 2|    |
  /                      ||                           /       2           /       2       |r |    |
 |                       ||                          /       y           /       y                |
 |    _________          ||                  2*r*   /   -1 + --    2*   /   -1 + --               |
 |   /  2    2           ||                        /          2        /          2               |
 | \/  r  - y   dy = C + |<                      \/          r       \/          r                |
 |                       ||                                                                       |
/                        ||                                    ________                           |
                         ||                                   /      2                            |
                         ||                                  /      y                             |
                         ||              2     /y\   r*y*   /   1 - --                            |
                         ||             r *asin|-|         /         2                            |
                         ||                    \r/       \/         r                             |
                         ||             ---------- + ------------------                otherwise  |
                         \\                 2                2                                    /
∫r2−y2 dy=C+{−ir2acosh⁡(yr)2−iry2−1+y2r2+iy32r−1+y2r2for ∣y2r2∣>1r2asin⁡(yr)2+ry1−y2r22otherwise\int \sqrt{r^{2} - y^{2}}\, dy = C + \begin{cases} - \frac{i r^{2} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{2} - \frac{i r y}{2 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{3}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{2} \operatorname{asin}{\left(\frac{y}{r} \right)}}{2} + \frac{r y \sqrt{1 - \frac{y^{2}}{r^{2}}}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
  1                                                                                                 
  /                                                                                                 
 |                                                                                                  
 |  /                             2                   4                     2              2        
 |  |        I*r               I*y                 I*y                 3*I*y              y         
 |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1   
 |  |        _________                3/2                 3/2             _________      | 2|       
 |  |       /       2        /      2\           /      2\               /       2       |r |       
 |  |      /       y         |     y |         3 |     y |              /       y                   
 |  |     /   -1 + --    2*r*|-1 + --|      2*r *|-1 + --|      2*r*   /   -1 + --                  
 |  |    /          2        |      2|           |      2|            /          2                  
 |  |  \/          r         \     r /           \     r /          \/          r                   
 |  |                                                                                               
 |  |                   ________                                                                    
 |  <                  /      2                                                                   dy
 |  |                 /      y                                                                      
 |  |           r*   /   1 - --                                                                     
 |  |               /         2                                2                                    
 |  |             \/         r            r                   y                                     
 |  |           ---------------- + ---------------- - ------------------              otherwise     
 |  |                  2                   ________             ________                            
 |  |                                     /      2             /      2                             
 |  |                                    /      y             /      y                              
 |  |                              2*   /   1 - --    2*r*   /   1 - --                             
 |  |                                  /         2          /         2                             
 |  \                                \/         r         \/         r                              
 |                                                                                                  
/                                                                                                   
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∫01{−ir−1+y2r2+3iy22r−1+y2r2+iy22r(−1+y2r2)32−iy42r3(−1+y2r2)32for y2∣r2∣>1r1−y2r22+r21−y2r2−y22r1−y2r2otherwise dy\int\limits_{0}^{1} \begin{cases} - \frac{i r}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{3 i y^{2}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{2}}{2 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{i y^{4}}{2 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{y^{2}}{\left|{r^{2}}\right|} > 1 \\\frac{r \sqrt{1 - \frac{y^{2}}{r^{2}}}}{2} + \frac{r}{2 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{y^{2}}{2 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy 
=
= 
  1                                                                                                 
  /                                                                                                 
 |                                                                                                  
 |  /                             2                   4                     2              2        
 |  |        I*r               I*y                 I*y                 3*I*y              y         
 |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1   
 |  |        _________                3/2                 3/2             _________      | 2|       
 |  |       /       2        /      2\           /      2\               /       2       |r |       
 |  |      /       y         |     y |         3 |     y |              /       y                   
 |  |     /   -1 + --    2*r*|-1 + --|      2*r *|-1 + --|      2*r*   /   -1 + --                  
 |  |    /          2        |      2|           |      2|            /          2                  
 |  |  \/          r         \     r /           \     r /          \/          r                   
 |  |                                                                                               
 |  |                   ________                                                                    
 |  <                  /      2                                                                   dy
 |  |                 /      y                                                                      
 |  |           r*   /   1 - --                                                                     
 |  |               /         2                                2                                    
 |  |             \/         r            r                   y                                     
 |  |           ---------------- + ---------------- - ------------------              otherwise     
 |  |                  2                   ________             ________                            
 |  |                                     /      2             /      2                             
 |  |                                    /      y             /      y                              
 |  |                              2*   /   1 - --    2*r*   /   1 - --                             
 |  |                                  /         2          /         2                             
 |  \                                \/         r         \/         r                              
 |                                                                                                  
/                                                                                                   
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∫01{−ir−1+y2r2+3iy22r−1+y2r2+iy22r(−1+y2r2)32−iy42r3(−1+y2r2)32for y2∣r2∣>1r1−y2r22+r21−y2r2−y22r1−y2r2otherwise dy\int\limits_{0}^{1} \begin{cases} - \frac{i r}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{3 i y^{2}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{2}}{2 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{i y^{4}}{2 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{y^{2}}{\left|{r^{2}}\right|} > 1 \\\frac{r \sqrt{1 - \frac{y^{2}}{r^{2}}}}{2} + \frac{r}{2 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{y^{2}}{2 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy 
Integral(Piecewise((-i*r/sqrt(-1 + y^2/r^2) + i*y^2/(2*r*(-1 + y^2/r^2)^(3/2)) - i*y^4/(2*r^3*(-1 + y^2/r^2)^(3/2)) + 3*i*y^2/(2*r*sqrt(-1 + y^2/r^2)), y^2/|r^2| > 1), (r*sqrt(1 - y^2/r^2)/2 + r/(2*sqrt(1 - y^2/r^2)) - y^2/(2*r*sqrt(1 - y^2/r^2)), True)), (y, 0, 1))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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