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The double integral of:
x^2+y^2+1
Plot:
x^2+y^2+1
x^2+y^2+1
Identical expressions
x^ two +y^ two + one
x squared plus y squared plus 1
x to the power of two plus y to the power of two plus one
x2+y2+1
x²+y²+1
x to the power of 2+y to the power of 2+1
x^2+y^2+1dx
Similar expressions
x^2-y^2+1
2*y/(x^2+y^2)+1
x^2+y^2-1

Integral of x^2+y^2+1 dy

The solution

You have entered [src]

       4                     
       /                     
      |                      
      |      / 2    2    \   
      |      \x  + y  + 1/ dy
      |                      
     /                       
   _________                 
  /       2                  
\/  16 - x

$$\int\limits_{\sqrt{16 - x^{2}}}^{4} \left(\left(x^{2} + y^{2}\right) + 1\right)\, dy$$

Integral(x^2 + y^2 + 1, (y, sqrt(16 - x^2), 4))

Detail solution

Integrate term-by-term:
1. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int x^{2}\, dy = x^{2} y$
  1. The integral of $y^{n}$ is $\frac{y^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int y^{2}\, dy = \frac{y^{3}}{3}$
  The result is: $x^{2} y + \frac{y^{3}}{3}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dy = y$
The result is: $x^{2} y + \frac{y^{3}}{3} + y$
Now simplify:

$y \left(x^{2} + \frac{y^{2}}{3} + 1\right)$
Add the constant of integration:

$y \left(x^{2} + \frac{y^{2}}{3} + 1\right)+ \mathrm{constant}$

The answer is:

$y \left(x^{2} + \frac{y^{2}}{3} + 1\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                             3       
 | / 2    2    \              y       2
 | \x  + y  + 1/ dy = C + y + -- + y*x 
 |                            3        
/

$$\int \left(\left(x^{2} + y^{2}\right) + 1\right)\, dy = C + x^{2} y + \frac{y^{3}}{3} + y$$

Simplify

The answer [src]

                     3/2                        
            /      2\         _________         
76      2   \16 - x /        /       2  /     2\
-- + 4*x  - ------------ - \/  16 - x  *\1 + x /
3                3

$$4 x^{2} - \frac{\left(16 - x^{2}\right)^{\frac{3}{2}}}{3} - \sqrt{16 - x^{2}} \left(x^{2} + 1\right) + \frac{76}{3}$$

                     3/2                        
            /      2\         _________         
76      2   \16 - x /        /       2  /     2\
-- + 4*x  - ------------ - \/  16 - x  *\1 + x /
3                3

$$4 x^{2} - \frac{\left(16 - x^{2}\right)^{\frac{3}{2}}}{3} - \sqrt{16 - x^{2}} \left(x^{2} + 1\right) + \frac{76}{3}$$

76/3 + 4*x^2 - (16 - x^2)^(3/2)/3 - sqrt(16 - x^2)*(1 + x^2)

Use the examples entering the upper and lower limits of integration.

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Integral of x^2+y^2+1 dy

Limits of integration:

The graph:

Piecewise:

The solution