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Identical expressions
twenty-four -2y-3z
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Integral of 24-2y-3z dz

The solution

You have entered [src]

     2*y                   
 8 - ---                   
      3                    
    /                      
   |                       
   |    (24 - 2*y - 3*z) dz
   |                       
  /                        
  0

$$\int\limits_{0}^{8 - \frac{2 y}{3}} \left(- 3 z + \left(24 - 2 y\right)\right)\, dz$$

Integral(24 - 2*y - 3*z, (z, 0, 8 - 2*y/3))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 3 z\right)\, dz = - 3 \int z\, dz$
  1. The integral of $z^{n}$ is $\frac{z^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int z\, dz = \frac{z^{2}}{2}$
  So, the result is: $- \frac{3 z^{2}}{2}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \left(24 - 2 y\right)\, dz = z \left(24 - 2 y\right)$
The result is: $- \frac{3 z^{2}}{2} + z \left(24 - 2 y\right)$
Now simplify:

$\frac{z \left(- 4 y - 3 z + 48\right)}{2}$
Add the constant of integration:

$\frac{z \left(- 4 y - 3 z + 48\right)}{2}+ \mathrm{constant}$

The answer is:

$\frac{z \left(- 4 y - 3 z + 48\right)}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                             2               
 |                           3*z                
 | (24 - 2*y - 3*z) dz = C - ---- + z*(24 - 2*y)
 |                            2                 
/

$$\int \left(- 3 z + \left(24 - 2 y\right)\right)\, dz = C - \frac{3 z^{2}}{2} + z \left(24 - 2 y\right)$$

Simplify

The answer [src]

             2                       
    /    2*y\                        
  3*|8 - ---|                        
    \     3 /    /    2*y\           
- ------------ + |8 - ---|*(24 - 2*y)
       2         \     3 /

$$- \frac{3 \left(8 - \frac{2 y}{3}\right)^{2}}{2} + \left(8 - \frac{2 y}{3}\right) \left(24 - 2 y\right)$$

             2                       
    /    2*y\                        
  3*|8 - ---|                        
    \     3 /    /    2*y\           
- ------------ + |8 - ---|*(24 - 2*y)
       2         \     3 /

$$- \frac{3 \left(8 - \frac{2 y}{3}\right)^{2}}{2} + \left(8 - \frac{2 y}{3}\right) \left(24 - 2 y\right)$$

-3*(8 - 2*y/3)^2/2 + (8 - 2*y/3)*(24 - 2*y)

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

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Integral of 24-2y-3z dz

Limits of integration:

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The solution