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Resolución de integrales/ sqrt(a^2-r^2)

Integral de sqrt(a^2-r^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
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 |     _________   
 |    /  2    2    
 |  \/  a  - r   dr
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∫01a2−r2 dr\int\limits_{0}^{1} \sqrt{a^{2} - r^{2}}\, dr 
Integral(sqrt(a^2 - r^2), (r, 0, 1))
Respuesta (Indefinida) [src] 
                         //     2      /r\                                                        \
                         ||  I*a *acosh|-|              3                                 | 2|    |
                         ||            \a/           I*r                 I*a*r            |r |    |
                         ||- ------------- + ------------------- - -----------------  for |--| > 1|
                         ||        2                   _________           _________      | 2|    |
  /                      ||                           /       2           /       2       |a |    |
 |                       ||                          /       r           /       r                |
 |    _________          ||                  2*a*   /   -1 + --    2*   /   -1 + --               |
 |   /  2    2           ||                        /          2        /          2               |
 | \/  a  - r   dr = C + |<                      \/          a       \/          a                |
 |                       ||                                                                       |
/                        ||                                    ________                           |
                         ||                                   /      2                            |
                         ||                                  /      r                             |
                         ||              2     /r\   a*r*   /   1 - --                            |
                         ||             a *asin|-|         /         2                            |
                         ||                    \a/       \/         a                             |
                         ||             ---------- + ------------------                otherwise  |
                         \\                 2                2                                    /
∫a2−r2 dr=C+{−ia2acosh⁡(ra)2−iar2−1+r2a2+ir32a−1+r2a2for ∣r2a2∣>1a2asin⁡(ra)2+ar1−r2a22otherwise\int \sqrt{a^{2} - r^{2}}\, dr = C + \begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left(\frac{r}{a} \right)}}{2} - \frac{i a r}{2 \sqrt{-1 + \frac{r^{2}}{a^{2}}}} + \frac{i r^{3}}{2 a \sqrt{-1 + \frac{r^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{r^{2}}{a^{2}}}\right| > 1 \\\frac{a^{2} \operatorname{asin}{\left(\frac{r}{a} \right)}}{2} + \frac{a r \sqrt{1 - \frac{r^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
  1                                                                                                 
  /                                                                                                 
 |                                                                                                  
 |  /                             2                   4                     2              2        
 |  |        I*a               I*r                 I*r                 3*I*r              r         
 |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1   
 |  |        _________                3/2                 3/2             _________      | 2|       
 |  |       /       2        /      2\           /      2\               /       2       |a |       
 |  |      /       r         |     r |         3 |     r |              /       r                   
 |  |     /   -1 + --    2*a*|-1 + --|      2*a *|-1 + --|      2*a*   /   -1 + --                  
 |  |    /          2        |      2|           |      2|            /          2                  
 |  |  \/          a         \     a /           \     a /          \/          a                   
 |  |                                                                                               
 |  |                   ________                                                                    
 |  <                  /      2                                                                   dr
 |  |                 /      r                                                                      
 |  |           a*   /   1 - --                                                                     
 |  |               /         2                                2                                    
 |  |             \/         a            a                   r                                     
 |  |           ---------------- + ---------------- - ------------------              otherwise     
 |  |                  2                   ________             ________                            
 |  |                                     /      2             /      2                             
 |  |                                    /      r             /      r                              
 |  |                              2*   /   1 - --    2*a*   /   1 - --                             
 |  |                                  /         2          /         2                             
 |  \                                \/         a         \/         a                              
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/                                                                                                   
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∫01{−ia−1+r2a2+3ir22a−1+r2a2+ir22a(−1+r2a2)32−ir42a3(−1+r2a2)32for r2∣a2∣>1a1−r2a22+a21−r2a2−r22a1−r2a2otherwise dr\int\limits_{0}^{1} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{r^{2}}{a^{2}}}} + \frac{3 i r^{2}}{2 a \sqrt{-1 + \frac{r^{2}}{a^{2}}}} + \frac{i r^{2}}{2 a \left(-1 + \frac{r^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i r^{4}}{2 a^{3} \left(-1 + \frac{r^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{r^{2}}{\left|{a^{2}}\right|} > 1 \\\frac{a \sqrt{1 - \frac{r^{2}}{a^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{r^{2}}{a^{2}}}} - \frac{r^{2}}{2 a \sqrt{1 - \frac{r^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dr 
=
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  1                                                                                                 
  /                                                                                                 
 |                                                                                                  
 |  /                             2                   4                     2              2        
 |  |        I*a               I*r                 I*r                 3*I*r              r         
 |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1   
 |  |        _________                3/2                 3/2             _________      | 2|       
 |  |       /       2        /      2\           /      2\               /       2       |a |       
 |  |      /       r         |     r |         3 |     r |              /       r                   
 |  |     /   -1 + --    2*a*|-1 + --|      2*a *|-1 + --|      2*a*   /   -1 + --                  
 |  |    /          2        |      2|           |      2|            /          2                  
 |  |  \/          a         \     a /           \     a /          \/          a                   
 |  |                                                                                               
 |  |                   ________                                                                    
 |  <                  /      2                                                                   dr
 |  |                 /      r                                                                      
 |  |           a*   /   1 - --                                                                     
 |  |               /         2                                2                                    
 |  |             \/         a            a                   r                                     
 |  |           ---------------- + ---------------- - ------------------              otherwise     
 |  |                  2                   ________             ________                            
 |  |                                     /      2             /      2                             
 |  |                                    /      r             /      r                              
 |  |                              2*   /   1 - --    2*a*   /   1 - --                             
 |  |                                  /         2          /         2                             
 |  \                                \/         a         \/         a                              
 |                                                                                                  
/                                                                                                   
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∫01{−ia−1+r2a2+3ir22a−1+r2a2+ir22a(−1+r2a2)32−ir42a3(−1+r2a2)32for r2∣a2∣>1a1−r2a22+a21−r2a2−r22a1−r2a2otherwise dr\int\limits_{0}^{1} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{r^{2}}{a^{2}}}} + \frac{3 i r^{2}}{2 a \sqrt{-1 + \frac{r^{2}}{a^{2}}}} + \frac{i r^{2}}{2 a \left(-1 + \frac{r^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i r^{4}}{2 a^{3} \left(-1 + \frac{r^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{r^{2}}{\left|{a^{2}}\right|} > 1 \\\frac{a \sqrt{1 - \frac{r^{2}}{a^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{r^{2}}{a^{2}}}} - \frac{r^{2}}{2 a \sqrt{1 - \frac{r^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dr 
Integral(Piecewise((-i*a/sqrt(-1 + r^2/a^2) + i*r^2/(2*a*(-1 + r^2/a^2)^(3/2)) - i*r^4/(2*a^3*(-1 + r^2/a^2)^(3/2)) + 3*i*r^2/(2*a*sqrt(-1 + r^2/a^2)), r^2/|a^2| > 1), (a*sqrt(1 - r^2/a^2)/2 + a/(2*sqrt(1 - r^2/a^2)) - r^2/(2*a*sqrt(1 - r^2/a^2)), True)), (r, 0, 1))

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

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